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Descriptive Statistics-II
Dr Mahmoud Alhussami
Shapes of Distribution A third important property of data – after location
and dispersion - is its shape Distributions of quantitative variables can be
described in terms of a number of features, many of which are related to the distributions’ physical appearance or shape when presented graphically. modality Symmetry and skewness Degree of skewness Kurtosis
Modality The modality of a distribution concerns
how many peaks or high points there are. A distribution with a single peak, one value
a high frequency is a unimodal distribution.
Modality A distribution with two
or more peaks called multimodal distribution.
Symmetry and Skewness A distribution is symmetric if the distribution could be split
down the middle to form two haves that are mirror images of one another.
In asymmetric distributions, the peaks are off center, with a bull of scores clustering at one end, and a tail trailing off at the other end. Such distributions are often describes as skewed. When the longer tail trails off to the right this is a positively
skewed distribution. E.g. annual income. When the longer tail trails off to the left this is called
negatively skewed distribution. E.g. age at death.
Symmetry and Skewness Shape can be described by degree of asymmetry (i.e.,
skewness). mean > median positive or right-skewness mean = median symmetric or zero-skewness mean < median negative or left-skewness
Positive skewness can arise when the mean is increased by some unusually high values.
Negative skewness can arise when the mean is decreased by some unusually low values.
Left skewed:
Right skewed:
Symmetric:
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April 13, 2023 8
Shapes of the DistributionShapes of the Distribution
Three common shapes of frequency Three common shapes of frequency distributions: distributions:
Symmetrical and bell shaped
Positively skewed or skewed to the right
Negatively skewed or skewed to the left
A B C
April 13, 2023 9
Shapes of the DistributionShapes of the Distribution
Three less common shapes of frequency Three less common shapes of frequency distributions: distributions:
Bimodal ReverseJ-shaped
Uniform
A B C
10
This guy took a VERY long time!
Degree of Skewness A skewness index can readily be calculated most
statistical computer program in conjunction with frequency distributions
The index has a value of 0 for perfectly symmetric distribution.
A positive value if there is a positive skew, and negative value if there is a negative skew.
A skewness index that is more than twice the value of its standard error can be interpreted as a departure from symmetry.
Measures of Skewness or Symmetry Pearson’s skewness coefficient
It is nonalgebraic and easily calculated. Also it is useful for quick estimates of symmetry .
It is defined as:skewness = mean-median/SD
Fisher’s measure of skewness. It is based on deviations from the mean to the
third power.
Pearson’s skewness coefficient
For a perfectly symmetrical distribution, the mean will equal the median, and the skewness coefficient will be zero. If the distribution is positively skewed the mean will be more than the median and the coefficient will be the positive. If the coefficient is negative, the distribution is negatively skewed and the mean less than the median.
Skewness values will fall between -1 and +1 SD units. Values falling outside this range indicate a substantially skewed distribution.
Hildebrand (1986) states that skewness values above 0.2 or below -0.2 indicate severe skewness.
Assumption of Normality
Many of the statistical methods that we will apply require the assumption that a variable or variables are normally distributed.
With multivariate statistics, the assumption is that the combination of variables follows a multivariate normal distribution.
Since there is not a direct test for multivariate normality, we generally test each variable individually and assume that they are multivariate normal if they are individually normal, though this is not necessarily the case.
Evaluating normality
There are both graphical and statistical methods for evaluating normality.
Graphical methods include the histogram and normality plot.
Statistical methods include diagnostic hypothesis tests for normality, and a rule of thumb that says a variable is reasonably close to normal if its skewness and kurtosis have values between –1.0 and +1.0.
None of the methods is absolutely definitive.
Transformations
When a variable is not normally distributed, we can create a transformed variable and test it for normality. If the transformed variable is normally distributed, we can substitute it in our analysis.
Three common transformations are: the logarithmic transformation, the square root transformation, and the inverse transformation.
All of these change the measuring scale on the horizontal axis of a histogram to produce a transformed variable that is mathematically equivalent to the original variable.
Types of Data Transformations for moderate skewness, use a square root
transformation. For substantial skewness, use a log
transformation. For sever skewness, use an inverse
transformation.
Computing “Explore” descriptive statistics
To compute the statistics needed for evaluating the normality of a variable, select the Explore… command from the Descriptive Statistics menu.
Adding the variable to be evaluated
First, click on the variable to be included in the analysis to highlight it.
Second, click on right arrow button to move the highlighted variable to the Dependent List.
Selecting statistics to be computed
To select the statistics for the output, click on the Statistics… command button.
Including descriptive statistics
First, click on the Descriptives checkbox to select it. Clear the other checkboxes.
Second, click on the Continue button to complete the request for statistics.
Selecting charts for the output
To select the diagnostic charts for the output, click on the Plots… command button.
Including diagnostic plots and statistics
First, click on the None option button on the Boxplots panel since boxplots are not as helpful as other charts in assessing normality.
Second, click on the Normality plots with tests checkbox to include normality plots and the hypothesis tests for normality.
Third, click on the Histogram checkbox to include a histogram in the output. You may want to examine the stem-and-leaf plot as well, though I find it less useful.
Finally, click on the Continue button to complete the request.
Completing the specifications for the analysis
Click on the OK button to complete the specifications for the analysis and request SPSS to produce the output.
TOTAL TIME SPENT ON THE INTERNET
100.0
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
HistogramF
requ
ency
50
40
30
20
10
0
Std. Dev = 15.35
Mean = 10.7
N = 93.00
The histogram
An initial impression of the normality of the distribution can be gained by examining the histogram.
In this example, the histogram shows a substantial violation of normality caused by a extremely large value in the distribution.
Normal Q-Q Plot of TOTAL TIME SPENT ON THE INTERNET
Observed Value
120100806040200-20-40
Exp
ecte
d N
orm
al
3
2
1
0
-1
-2
-3
The normality plot
The problem with the normality of this variable’s distribution is reinforced by the normality plot.
If the variable were normally distributed, the red dots would fit the green line very closely. In this case, the red points in the upper right of the chart indicate the severe skewing caused by the extremely large data values.
Tests of Normality
.246 93 .000 .606 93 .000TOTAL TIME SPENTON THE INTERNET
Statistic df Sig. Statistic df Sig.
Kolmogorov-Smirnova
Shapiro-Wilk
Lilliefors Significance Correctiona.
The test of normality
Problem 1 asks about the results of the test of normality. Since the sample size is larger than 50, we use the Kolmogorov-Smirnov test. If the sample size were 50 or less, we would use the Shapiro-Wilk statistic instead.
The null hypothesis for the test of normality states that the actual distribution of the variable is equal to the expected distribution, i.e., the variable is normally distributed. Since the probability associated with the test of normality is < 0.001 is less than or equal to the level of significance (0.01), we reject the null hypothesis and conclude that total hours spent on the Internet is not normally distributed. (Note: we report the probability as <0.001 instead of .000 to be clear that the probability is not really zero.)
The answer to problem 1 is false.
The assumption of normality script
An SPSS script to produce all of the output that we have produced manually is available on the course web site.
After downloading the script, run it to test the assumption of linearity.
Select Run Script… from the Utilities menu.
Selecting the assumption of normality script
First, navigate to the folder containing your scripts and highlight the NormalityAssumptionAndTransformations.SBS script.
Second, click on the Run button to activate the script.
Specifications for normality script
The default output is to do all of the transformations of the variable. To exclude some transformations from the calculations, clear the checkboxes.
Third, click on the OK button to run the script.
First, move variables from the list of variables in the data set to the Variables to Test list box.
Tests of Normality
.246 93 .000 .606 93 .000TOTAL TIME SPENTON THE INTERNET
Statistic df Sig. Statistic df Sig.
Kolmogorov-Smirnova
Shapiro-Wilk
Lilliefors Significance Correctiona.
The test of normality
The script produces the same output that we computed manually, in this example, the tests of normality.
When transformations do not work When none of the transformations induces
normality in a variable, including that variable in the analysis will reduce our effectiveness at identifying statistical relationships, i.e. we lose power.
We do have the option of changing the way the information in the variable is represented, e.g. substitute several dichotomous variables for a single metric variable.
Fisher’s Measure of Skewness The formula for Fisher’s skewness statistic is based on
deviations from the mean to the third power. The measure of skewness can be interpreted in terms of
the normal curve A symmetrical curve will result in a value of 0. If the skewness value is positive, them the curve is skewed to
the right, and vice versa for a distribution skewed to the left. A z-score is calculated by dividing the measure of skewness
by the standard error for skewness. Values above +1.96 or below -1.96 are significant at the 0.05 level because 95% of the scores in a normal deviation fall between +1.96 and -1.96 from the mean.
E.g. if Fisher’s skewness= 0.195 and st.err. =0.197 the z-score = 0.195/0.197 = 0.99
Kurtosis The distribution’s kurtosis is concerns how
pointed or flat its peak. Two types:
Leptokurtic distribution (mean thin). Platykurtic distribution (means flat).
Kurtosis There is a statistical index of kurtosis that can be
computed when computer programs are instructed to produce a frequency distribution
For kurtosis index, a value of zero indicates a shape that is neither flat nor pointed.
Positive values on the kurtosis statistics indicate greater peakedness, and negative values indicate greater flatness.
Fishers’ measure of Kurtosis Fisher’s measure is based on deviation
from the mean to the fourth power. A z-score is calculated by dividing the
measure of kurtosis by the standard error for kurtosis.
Descriptives
10.731 1.5918
7.570
13.893
8.295
5.500
235.655
15.3511
.2
102.0
101.8
10.200
3.532 .250
15.614 .495
Mean
Lower Bound
Upper Bound
95% ConfidenceInterval for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
TOTAL TIME SPENTON THE INTERNET
Statistic Std. Error
Table of descriptive statistics
To answer problem 2, we look at the values for skewness and kurtosis in the Descriptives table.
The skewness and kurtosis for the variable both exceed the rule of thumb criteria of 1.0. The variable is not normally distributed.
The answer to problem 2 if false.
Other problems on assumption of normality
A problem may ask about the assumption of normality for a nominal level variable. The answer will be “An inappropriate application of a statistic” since there is no expectation that a nominal variable be normal.
A problem may ask about the assumption of normality for an ordinal level variable. If the variable or transformed variable is normal, the correct answer to the question is “True with caution” since we may be required to defend treating an ordinal variable as metric.
Questions will specify a level of significance to use and the statistical evidence upon which you should base your answer.
Normal Distribution Also called belt shaped curve, normal
curve, or Gaussian distribution. A normal distribution is one that is
unimodal, symmetric, and not too peaked or flat.
Given its name by the French mathematician Quetelet who, in the early 19th century noted that many human attributes, e.g. height, weight, intelligence appeared to be distributed normally.
April 13, 2023 40
Normal DistributionNormal Distribution The normal curve is unimodal and symmetric The normal curve is unimodal and symmetric
about its mean (about its mean (). ). In this distribution the mean, median and mode In this distribution the mean, median and mode
are all identical.are all identical. The standard deviation (The standard deviation () specifies the amount ) specifies the amount
of dispersion around the mean.of dispersion around the mean. The two parameters The two parameters and and completely define a completely define a
normal curve.normal curve.
Also called a Probability density function. The probability is interpreted as "area under the curve."
The random variable takes on an infinite # of values within a given interval
The probability that X = any particular value is 0. Consequently, we talk about intervals. The probability is = to the area under the curve.
The area under the whole curve = 1.
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Normal Distribution
X is the random variable.μ is the mean value.σ is the standard deviation (std) value.e = 2.7182818... constant.π = 3.1415926... constant.
Importance of Normal Distribution to Statistics Although most distributions are not
exactly normal, most variables tend to have approximately normal distribution.
Many inferential statistics assume that the populations are distributed normally.
The normal curve is a probability distribution and is used to answer questions about the likelihood of getting various particular outcomes when sampling from a population.
Probabilities are obtained by getting the area under the curve inside of a particular interval. The area under the curve = the proportion of times under identical (repeated) conditions that a particular range of values will occur.
Characteristics of the Normal distribution: It is symmetric about the mean μ. Mean = median = mode. [“bell-shaped” curve] f(X) decreases as X gets farther and farther away from
the mean. It approaches horizontal axis asymptotically:- ∞ < X < + ∞. This means that there is always some probability (area) for extreme values.
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Why Do We Like The Normal Why Do We Like The Normal Distribution So MuchDistribution So Much??
There is nothing “special” about standard There is nothing “special” about standard normal scoresnormal scores These can be computed for observations from any These can be computed for observations from any
sample/population of continuous data valuessample/population of continuous data values The score measures how far an observation is from The score measures how far an observation is from
its mean in standard units of statistical distanceits mean in standard units of statistical distance But, if distribution is not normal, we may not be But, if distribution is not normal, we may not be
able to use Z-score approach.able to use Z-score approach.
April 13, 2023 47
Probability DistributionsProbability Distributions Any characteristic that can be measured or Any characteristic that can be measured or
categorized is called a categorized is called a variablevariable.. If the variable can assume a number of different If the variable can assume a number of different
values such that any particular outcome is values such that any particular outcome is determined by chance it is called adetermined by chance it is called a random random variable.variable.
Every random variable has a corresponding Every random variable has a corresponding probability distributionprobability distribution..
The probability distribution applies the theory of The probability distribution applies the theory of probability to describe the behavior of the probability to describe the behavior of the random variablerandom variable..
April 13, 2023 48
Discrete Probability Discrete Probability DistributionsDistributions Binomial distribution – the random variable Binomial distribution – the random variable
can only assume 1 of 2 possible outcomes. can only assume 1 of 2 possible outcomes. There are a fixed number of trials and the There are a fixed number of trials and the results of the trials are independent.results of the trials are independent.
i.e. flipping a coin and counting the number of heads in i.e. flipping a coin and counting the number of heads in 10 trials.10 trials.
Poisson Distribution – random variable can Poisson Distribution – random variable can assume a value between 0 and infinity.assume a value between 0 and infinity.
Counts usually follow a Poisson distribution (i.e. number Counts usually follow a Poisson distribution (i.e. number of ambulances needed in a city in a given night) of ambulances needed in a city in a given night)
April 13, 2023 49
Discrete Random VariableDiscrete Random Variable A A discrete random variablediscrete random variable X has a finite number of possible X has a finite number of possible
values. The values. The probability distributionprobability distribution of X lists the values and of X lists the values and their probabilities.their probabilities.
1.1. Every probability pEvery probability pii is a number between 0 and 1. is a number between 0 and 1.
2.2. The sum of the probabilities must be 1.The sum of the probabilities must be 1. Find the probabilities of any event by adding the probabilities Find the probabilities of any event by adding the probabilities
of the particular values that make up the event.of the particular values that make up the event.
Value of XValue of Xxx11xx22xx33……xxkk
ProbabilityProbabilitypp11pp22pp33……ppkk
April 13, 2023 50
ExampleExample The instructor in a large class gives 15% each of A’s and D’s, The instructor in a large class gives 15% each of A’s and D’s,
30% each of B’s and C’s and 10% F’s. The student’s grade on 30% each of B’s and C’s and 10% F’s. The student’s grade on a 4-point scale is a random variable X (A=4).a 4-point scale is a random variable X (A=4).
What is the probability that a student selected at random will What is the probability that a student selected at random will have a B or better?have a B or better?
ANSWER: P (grade of 3 or 4)=P(X=3) + P(X=4)ANSWER: P (grade of 3 or 4)=P(X=3) + P(X=4)
= 0.3 + 0.15 = 0.45= 0.3 + 0.15 = 0.45
GradeGradeF=0F=0D=1D=1C=2C=2B=3B=3A=4A=4
ProbabilityProbability0.100.10..1515..3030..3030..1515
April 13, 2023 51
Continuous Probability Continuous Probability DistributionsDistributions When it follows a Binomial or a Poisson When it follows a Binomial or a Poisson
distribution the variable is restricted to taking on distribution the variable is restricted to taking on integer values only.integer values only.
Between two values of a continuous random Between two values of a continuous random variable we can always find a third.variable we can always find a third.
A histogram is used to represent a discrete A histogram is used to represent a discrete probability distribution and a smooth curve called probability distribution and a smooth curve called the the probability density probability density is used to represent a is used to represent a continuous probability distribution.continuous probability distribution.
April 13, 2023 52
Normal DistributionNormal Distribution
Q Is every variable normally distributed?Is every variable normally distributed?A Absolutely notAbsolutely notQ Then why do we spend so much time Then why do we spend so much time
studying the normal distribution?studying the normal distribution?A Some variables are normally distributed; Some variables are normally distributed;
a bigger reason is the “Central Limit a bigger reason is the “Central Limit Theorem”!!!!!!!!!!!!!!!!!!!!!!!!!!!?????????Theorem”!!!!!!!!!!!!!!!!!!!!!!!!!!!?????????????
Central Limit TheoremCentral Limit Theorem describes the characteristics of the "population of the
means" which has been created from the means of an infinite number of random population samples of size (N), all of them drawn from a given "parent population".
It predicts that regardless of the distribution of the parent population: The mean of the population of means is always equal to the
mean of the parent population from which the population samples were drawn.
The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N).
The distribution of means will increasingly approximate a normal distribution as the size N of samples increases.
Central Limit TheoremCentral Limit Theorem A consequence of Central Limit Theorem is that if we
average measurements of a particular quantity, the distribution of our average tends toward a normal one.
In addition, if a measured variable is actually a combination of several other uncorrelated variables, all of them "contaminated" with a random error of any distribution, our measurements tend to be contaminated with a random error that is normally distributed as the number of these variables increases.
Thus, the Central Limit Theorem explains the ubiquity of the famous bell-shaped "Normal distribution" (or "Gaussian distribution") in the measurements domain.
Note that the normal distribution is defined by two parameters, μ and σ . You can draw a normal distribution for any μ and σ combination. There is one normal distribution, Z, that is special. It has a μ = 0 and a σ = 1. This is the Z distribution, also called the standard normal distribution. It is one of trillions of normal distributions we could have selected.
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April 13, 2023 56
Standard Normal VariableStandard Normal Variable It is customary to call a standard normal random It is customary to call a standard normal random
variable Z.variable Z. The outcomes of the random variable Z are The outcomes of the random variable Z are
denoted by denoted by z.z. The table in the coming slide give the area under The table in the coming slide give the area under
the curve (probabilities) between the mean and the curve (probabilities) between the mean and z.z. The probabilities in the table refer to the The probabilities in the table refer to the
likelihood that a randomly selected value Z is likelihood that a randomly selected value Z is equal to or less than a given value of equal to or less than a given value of zz and and greater than 0 (the mean of the standard greater than 0 (the mean of the standard normal).normal).
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Source: Levine et al, Business Statistics, Pearson.
April 13, 2023 58
The 68-95-99.7 Rule for the The 68-95-99.7 Rule for the Normal DistributionNormal Distribution 68% of the observations fall within one 68% of the observations fall within one
standard deviation of the meanstandard deviation of the mean 95% of the observations fall within two 95% of the observations fall within two
standard deviations of the meanstandard deviations of the mean 99.7% of the observations fall within three 99.7% of the observations fall within three
standard deviations of the meanstandard deviations of the mean When applied to ‘real data’, these When applied to ‘real data’, these
estimates are considered approximate!estimates are considered approximate!
Remember these probabilities (percentages):
Practice: Find these values yourself using the Z table.
Two Sample Z Test59
#standard deviations from the mean
Approx. area under the normal curve
±1.68
±1.645.90
±1.96.95
±2.955
±2.575.99
±3.997
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Standard Normal CurveStandard Normal Curve
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Standard Normal Standard Normal DistributionDistribution
50% of probability in here–probability=0.5
50% of probability in here –probability=0.5
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Standard Normal Standard Normal DistributionDistribution
2.5% of probability in here
2.5% of probability in here
95% of probability in here
Standard Normal Distribution with 95% area
marked
April 13, 2023 63
Calculating ProbabilitiesCalculating Probabilities Probability calculations are always Probability calculations are always
concerned with finding the probability that concerned with finding the probability that the variable assumes any value in an the variable assumes any value in an interval between two specific points interval between two specific points a a andand b.b.
The probability that a continuous variable The probability that a continuous variable assumes the a value between assumes the a value between a a andand b b is is the area under the graph of the density the area under the graph of the density betweenbetween a a and and b.b.
Normal Distribution 64
If the weight of males is N.D. with μ=150 and σ=10, what is the probability that a randomly selected male will weigh between 140 lbs and 155 lbs?[Important Note: Always remember that the probability that X is equal to any one particular value is zero, P(X=value) =0, since the normal distribution is continuous.]
Solution:
Z = (140 – 150)/ 10 = -1.00 s.d. from meanArea under the curve = .3413 (from Z table)
Z = (155 – 150) / 10 =+.50 s.d. from meanArea under the curve = .1915 (from Z table)
Answer: .3413 + .1915 = .5328
65
150 155
X
Z 0.5 -1 0
140
Example For example: What’s the probability of getting a math SAT score of 575 or less, =500 and =50?
5.150
500575
Z
i.e., A score of 575 is 1.5 standard deviations above the mean
5.1
2
1575
200
)50
500(
2
1 22
2
1
2)50(
1)575( dzedxeXP
Zx
Yikes! But to look up Z= 1.5 in standard normal chart (or enter into SAS) no problem! = .9332
If IQ is ND with a mean of 100 and a S.D. of 10, what percentage of the population will have (a)IQs ranging from 90 to 110? (b)IQs ranging from 80 to 120?Solution: Z = (90 – 100)/10 = -1.00Z = (110 -100)/ 10 = +1.00 Area between 0 and 1.00 in the Z-table is .3413; Area between 0 and -1.00 is also .3413 (Z-distribution is symmetric). Answer to part (a) is .3413 + .3413 = .6826.
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(b) IQs ranging from 80 to 120?Solution: Z = (80 – 100)/10 = -2.00Z = (120 -100)/ 10 = +2.00 Area between =0 and 2.00 in the Z-table is
.4772; Area between 0 and -2.00 is also .4772 (Z-distribution is symmetric).
Answer is .4772 + .4772 = .9544.
68
Suppose that the average salary of college graduates is N.D. with μ=$40,000 and σ=$10,000.
(a) What proportion of college graduates will earn $24,800 or less?
(b) What proportion of college graduates will earn $53,500 or more?
(c) What proportion of college graduates will earn between $45,000 and $57,000?
(d) Calculate the 80th percentile. (e) Calculate the 27th percentile.
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(a) What proportion of college graduates will earn $24,800 or less?Solution: Convert the $24,800 to a Z-score: Z = ($24,800 - $40,000)/$10,000 = -1.52. Always DRAW a picture of the distribution to help you solve these problems.
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First Find the area between 0 and -1.52 in the Z-table. From the Z table, that area is .4357. Then, the area from -1.52 to - ∞ is .5000 - .4357 = .0643.Answer: 6.43% of college graduates will earn less than $24,800.
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$24,800 $40,000
-1.52 0
.4357
X
Z
(b) What proportion of college graduates will earn $53,500 or more?Solution: Convert the $53,500 to a Z-score.Z = ($53,500 - $40,000)/$10,000 = +1.35. Find the area between 0 and +1.35 in the Z-table: .4115 is the table value.When you DRAW A PICTURE (above) you see that you need the area in the tail: .5 - .4115 - .0885.Answer: .0885. Thus, 8.85% of college graduates will earn $53,500 or more.
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$40,000 $53,500
Z
0 +1.35
.4115
.0885
(c) What proportion of college graduates will earn between $45,000 and $57,000?
Z = $45,000 – $40,000 / $10,000 = .50Z = $57,000 – $40,000 / $10,000 = 1.70
From the table, we can get the area under the curve between the mean (0) and .5; we can get the area between 0 and 1.7. From the picture we see that neither one is what we need.What do we do here? Subtract the small piece from the big piece to get exactly what we need.Answer: .4554 − .1915 = .2639
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$40k
Z
0 1.7
$45k $57k
.5
.19
15
.4554
Parts (d) and (e) of this example ask you to compute percentiles. Every Z-score is associated with a percentile. A Z-score of 0 is the 50th percentile. This means that if you take any test that is normally distributed (e.g., the SAT exam), and your Z-score on the test is 0, this means you scored at the 50th percentile. In fact, your score is the mean, median, and mode.
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(d) Calculate the 80th percentile.
Solution:First, what Z-score is associated with the 80th percentile? A Z-score of approximately +.84 will give you about .3000 of the area under the curve. Also, the area under the curve between -∞ and 0 is .5000. Therefore, a Z-score of +.84 is associated with the 80th percentile.
Now to find the salary (X) at the 80th percentile: Just solve for X: +.84 = (X−$40,000)/$10,000 X = $40,000 + $8,400 = $48,400.
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$40,000
Z
0 .84
.3000.5000
ANSWER
(e) Calculate the 27th percentile.
Solution: First, what Z-score is associated with the 27th percentile? A Z-score of approximately -.61will give you about .2300 of the area under the curve, with .2700 in the tail. (The area under the curve between 0 and -.61 is .2291 which we are rounding to .2300). Also, the area under the curve between 0 and ∞ is .5000. Therefore, a Z-score of -.61 is associated with the 27th percentile.
Now to find the salary (X) at the 27th percentile: Just solve for X: -0.61 =(X−$40,000)/$10,000 X = $40,000 - $6,100 = $33,900
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$40,000
Z
0-.61
.5000.2300
ANSWER
.2700
April 13, 2023 77
T-DistributionT-Distribution Similar to the standard normal in that it is unimodal, bell-Similar to the standard normal in that it is unimodal, bell-
shaped and symmetric.shaped and symmetric. The tail on the distribution are “thicker” than the standard The tail on the distribution are “thicker” than the standard
normalnormal The distribution is indexed by “degrees of freedom” (df).The distribution is indexed by “degrees of freedom” (df). The degrees of freedom measure the amount of information The degrees of freedom measure the amount of information
available in the data set that can be used for estimating the available in the data set that can be used for estimating the population variance (df=n-1).population variance (df=n-1).
Area under the curve still equals 1.Area under the curve still equals 1. Probabilities for the t-distribution with infinite df equals those Probabilities for the t-distribution with infinite df equals those
of the standard normal.of the standard normal.
April 13, 2023 78
T-DistributionT-Distribution
The table of t-distribution will give you the The table of t-distribution will give you the probability to the right of a critical value – probability to the right of a critical value – i.e. area in the upper tail.i.e. area in the upper tail.
We are only given the area (or probability) We are only given the area (or probability) for a few selected critical values for each for a few selected critical values for each degree of freedom.degree of freedom.
April 13, 2023 79
T-Distribution ExampleT-Distribution Example
For a t-curve from a sample of size 15 find For a t-curve from a sample of size 15 find the area to the left of 2.145.the area to the left of 2.145.
Answer: df=15-1=14Answer: df=15-1=14 In the table of the t~distribution, the area to In the table of the t~distribution, the area to
the right of 2.145 is 0.025.the right of 2.145 is 0.025. Therefore the area to the left of 2.145 is: Therefore the area to the left of 2.145 is:
1-0.025=0.9751-0.025=0.975
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Graphical MethodsGraphical Methods
Frequency DistributionFrequency Distribution HistogramHistogram Frequency PolygonFrequency Polygon Cumulative Frequency GraphCumulative Frequency Graph Pie Chart.
Presenting Data Table
Condenses data into a form that can make them easier to understand;
Shows many details in summary fashion;BUT
Since table shows only numbers, it may not be readily understood without comparing it to other values.
Principles of Table Construction Don’t try to do too much in a table Us white space effectively to make table
layout pleasing to the eye. Make sure tables & test refer to each
other. Use some aspect of the table to order &
group rows & columns.
Principles of Table Construction If appropriate, frame table with summary
statistics in rows & columns to provide a standard of comparison.
Round numbers in table to one or two decimal places to make them easily understood.
When creating tables for publication in a manuscript, double-space them unless contraindicated by journal.
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Frequency DistributionsFrequency Distributions
A useful way to present data when you A useful way to present data when you have a large data set is the formation of a have a large data set is the formation of a frequency tablefrequency table or or frequency frequency distribution.distribution.
FrequencyFrequency – the number of observations – the number of observations that fall within a certain range of the data.that fall within a certain range of the data.
85
Frequency TableFrequency Table
AgeAgeNumber of DeathsNumber of Deaths
<<11564564
1-41-48686
5-145-14127127
15-2415-24490490
25-3425-346666
35-4435-44806806
45-5445-541,4251,425
55-6455-643,5113,511
65-7465-746,9326,932
75-8475-8410,10110,101
8585++98259825
TotalTotal34,52434,524
86
Frequency TableFrequency Table
Data Data IntervalsIntervals
FrequencyFrequencyCumulative Cumulative FrequencyFrequency
Relative Relative FrequencyFrequency
)%()%(
Cumulative Cumulative Relative Relative FrequencyFrequency)%( )%(
10-1910-195555
20-2920-2918182323
30-3930-3910103333
40-4940-4913134646
50-5950-59445050
60-6960-69445454
70-7970-79225656
TotalTotal
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Cumulative Relative Cumulative Relative FrequencyFrequency Cumulative Relative FrequencyCumulative Relative Frequency – the – the
percentage of persons having a percentage of persons having a measurement less than or equal to the measurement less than or equal to the upper boundary of the class interval.upper boundary of the class interval. i.e. cumulative relative frequency for the 3i.e. cumulative relative frequency for the 3rdrd
interval of our data example:interval of our data example: 8.8+13.3+17.5 = 59.6%8.8+13.3+17.5 = 59.6%
- We say that 59.6% of the children have weights below - We say that 59.6% of the children have weights below 39.5 pounds.39.5 pounds.
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Number of IntervalsNumber of Intervals
There is no clear-cut rule on the number of There is no clear-cut rule on the number of intervals or classes that should be used.intervals or classes that should be used.
Too many intervals – the data may not be Too many intervals – the data may not be summarized enough for a clear summarized enough for a clear visualization of how they are distributed.visualization of how they are distributed.
Too few intervals – the data may be over-Too few intervals – the data may be over-summarized and some of the details of the summarized and some of the details of the distribution may be lost.distribution may be lost.
Presenting DataChart
- Visual representation of a frequency distribution that helps to
gain insight about what the data mean. - Built with lines, area & text: bar charts Ex: bar chart, pie chart
Bar Chart Simplest form of chart Used to display
nominal or ordinal data
ETHICAL ISSUES SCALE
ITEM 8
ACTING AGAINST YOUR OWN PERSONAL/RELIGIOUS VIEWS
FrequentlySomet imesSeldomNeverP
ER
CE
NT
60
50
40
30
20
10
0
Horizontal Bar Chart
CLINICAL PRACTICE AREAC
LIN
ICA
L P
RA
CT
ICE
AR
EA
Acute CareCritical CareGerontology
Post AnesthesiaPerinatal
Clinical ResearchFamily Nursing
NeonatalPsych/Mental Health
Community HealthGeneral Practice
OrthopedicsPrimary Care
Operating RoomMedical
OncologyOther
PERCENT
14121086420
Cluster Bar Chart
RN HIGHEST EDUCATION
Post Bac
Bachelor Degree
Associate Degree
Diploma
PE
RC
EN
T
70
60
50
40
30
20
10
0
Employment
Full time RN
Part time RN
Self employed
Pie Chart Alternative to bar
chart Circle partitioned into
percentage distributions of qualitative variables with total area of 100%
Doctorate NonNursing
Doctorate Nursing
MS NonNursing
MS Nursing
Juris Doctor
BS NonNursing
BS Nursing
AD Nursing
Diploma-Nursing
Missing
Histogram Appropriate for interval, ratio and
sometimes ordinal data Similar to bar charts but bars are placed
side by side Often used to represent both frequencies
and percentages Most histograms have from 5 to 20 bars
Histogram
SF-36 VITALITY SCORES
100.0
90.0
80.0
70.0
60.0
50.0
40.0
30.0
20.0
10.0
0.0
FR
EQ
UE
NC
Y
80
60
40
20
0
Std. Dev = 22.17
Mean = 61.6
N = 439.00
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Pictures of Data: HistogramsPictures of Data: HistogramsBlood pressure data on a sample of 113 menBlood pressure data on a sample of 113 men
Histogram of the Systolic Blood Pressure for 113 men. Each bar spans a width of 5 mmHg on the horizontal axis. The height of each bar represents the number of individuals with SBP in that range.
05
1015
20N
um
be
r o
f M
en
80 100 120 140 160Systolic BP (mmHg)
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Frequency PolygonFrequency Polygon
•First place a dot at the midpoint of the upper base of each rectangular bar.
•The points are connected with straight lines.
•At the ends, the points are connected to the midpoints of the previous and succeeding intervals (these intervals have zero frequency).
Frequency Polygon
0
2
4
6
8
10
12
14
16
18
20
4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5
Childrens w eights
Hallmarks of a Good Chart Simple & easy to read Placed correctly within text Use color only when it has a purpose, not
solely for decoration Make sure others can understand chart;
try it out on somebody first Remember: A poor chart is worse than no
chart at all.
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Cumulative Frequency PlotCumulative Frequency PlotWeights of Daycare Children
0%
20%
40%
60%
80%
100%
120%
9.5 19.5 29.5 39.5 49.5 59.5 69.5 79.5 89.5
Weight Range
Per
cen
t o
f C
hild
ren
•Place a point with a horizontal axis marked at the upper class boundary and a vertical axis marked at the corresponding cumulative frequency.
•Each point represents the cumulative relative frequency and the points are connected with straight lines.
•The left end is connected to the lower boundary of the first interval that has data.
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Coefficient of CorrelationCoefficient of Correlation
Measure of linear association between 2 Measure of linear association between 2 continuous variables.continuous variables.
Setting:Setting: two measurements are made for each two measurements are made for each
observation.observation. Sample consists of pairs of values and you Sample consists of pairs of values and you
want to determine the association between the want to determine the association between the variables.variables.
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Association ExamplesAssociation Examples Example 1: Association between a mother’s Example 1: Association between a mother’s
weight and the birth weight of her childweight and the birth weight of her child 2 measurements: mother’s weight and baby’s weight2 measurements: mother’s weight and baby’s weight
Both continuous measures Both continuous measures
Example 2: Association between a risk factor and Example 2: Association between a risk factor and a diseasea disease 2 measurements: disease status and risk factor status2 measurements: disease status and risk factor status
Both dichotomous measurementsBoth dichotomous measurements
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Correlation AnalysisCorrelation Analysis
When you have 2 continuous When you have 2 continuous measurements you use correlation measurements you use correlation analysis to determine the relationship analysis to determine the relationship between the variables.between the variables.
Through correlation analysis you can Through correlation analysis you can calculate a number that relates to the calculate a number that relates to the strength of the linear association.strength of the linear association.
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Types of RelationshipsTypes of Relationships
There are 2 types of relationships:There are 2 types of relationships: Deterministic relationshipDeterministic relationship – the values of the 2 – the values of the 2
variables are related through an exact variables are related through an exact mathematical formula. mathematical formula.
Statistical relationship Statistical relationship – this is not a – this is not a perfectperfect
relationship!!! relationship!!!
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Scatter Plots and Scatter Plots and AssociationAssociation You can plot the 2 variables in a scatter plot (one You can plot the 2 variables in a scatter plot (one
of the types of charts in SPSS/Excel).of the types of charts in SPSS/Excel). The pattern of the “dots” in the plot indicate the The pattern of the “dots” in the plot indicate the
statistical relationshipstatistical relationship between the variables (the between the variables (the strength and the direction).strength and the direction). Positive relationship – pattern goes from lower left to Positive relationship – pattern goes from lower left to
upper right.upper right. Negative relationship – pattern goes from upper left to Negative relationship – pattern goes from upper left to
lower right.lower right. The more the dots cluster around a straight line the The more the dots cluster around a straight line the
stronger the linear relationship.stronger the linear relationship.
105
Birth Weight DataBirth Weight Data
x (oz) y(%)112 63111 66107 72119 5292 7580 11881 12084 114
118 42106 72103 9094 91
x – birth weight in ounces
y – increase in weight between 70th and 100th days of life, expressed as a percentage of birth weight
April 13, 2023 106
Pearson Correlation Pearson Correlation CoefficientCoefficient
Birth Weight Data
40
50
60
70
80
90
100
110
120
70 80 90 100 110 120 130 140
Birth Weight )in ounces(
Incr
ease
in
Bir
th W
eig
ht
)%(
April 13, 2023 107
Calculations of Correlation Calculations of Correlation CoefficientCoefficient In SPSS:In SPSS:
Go to TOOLS menu and select DATA ANALYSIS.Go to TOOLS menu and select DATA ANALYSIS. Highlight CORRELATION and click “ok”Highlight CORRELATION and click “ok” Enter INPUT RANGE (2 columns of data that Enter INPUT RANGE (2 columns of data that
contain “x” and “y”)contain “x” and “y”) Click “ok” (cells where you want the answer to Click “ok” (cells where you want the answer to
be placed.be placed.
April 13, 2023 108
Pearson Correlation ResultsPearson Correlation Results
x (oz) y(%)
x (oz) 1y(%) -0.94629 1
Pearson Correlation Coefficient = -0.946
Interpretation:
- values near 1 indicate strong positive linear relationship
- values near –1 indicate strong negative linear relationship
- values near 0 indicate a weak linear association
April 13, 2023 109
CAUTIONCAUTION!!!!!!!!
Interpreting the correlation coefficient Interpreting the correlation coefficient should be done cautiously!should be done cautiously!
A result of 0 does not mean there is NO A result of 0 does not mean there is NO relationship …. It means there is no relationship …. It means there is no linearlinear association.association.
There may be a perfect non-linear There may be a perfect non-linear association.association.
The Uses of Frequency Distributions Becoming familiar with dataset. Cleaning the data.
Outliers-values that lie outside the normal range of values for other cases.
Inspecting the data for missing values. Testing assumptions for statistical tests.
Assumption is a condition that is presumed to be true and when ignored or violated can lead to misleading or invalid results.
When DV is not normally distributed researchers have to choose between three options:
Select a statistical test that does not assume a normal distribution. Ignore the violation of the assumption. Transform the variable to better approximate a distribution that is
normal. Please consult the various data transformation.
The Uses of Frequency Distributions Obtaining information about sample
characteristics. Directing answering research questions.
Outliers Are values that are extreme relative to the bulk of
scores in the distribution. They appear to be inconsistent with the rest of
the data. Advantages:
They may indicate characteristics of the population that would not be known in the normal course of analysis.
Disadvantages: They do not represent the population Run counter to the objectives of the analysis Can distort statistical tests.
Sources of Outliers An error in the recording of the data. A failure of data collection, such as not
following sample criteria (e.g. inadvertently admitting a disoriented patient into a study), a subject not following instructions on a questionnaire, or equipment failure.
An actual extreme value from an unusual subjects.
Methods to Identify Outliers Traditional way of labeling outliers, any
value more than 3SD from the mean. Values that are more than 3 IQRs from the
upper or lower edge of the box plot are extreme outliers.
Values between 1.5 and 3 IQRs from the upper and lower edges of the box are minor outliers.
Handling Outliers Analyze the data two ways:
With the outliers in the distribution With outliers removed.
If the results are similar, as they are likely to be if the sample size is large, then the outliers may be ignored.
If the results are not similar, then a statistical analysis that is resistant to outliers can be used (e.g. median and IQR).
If you want to use a mean with outliers, then the trimmed mean is an option. If calculated with a certain percentage of the extreme values removed from both ends of the distribution (e.g. n=100, then 5% trimmed mean is the mean of the middle 90% of the observation).
Handling Outliers Another alternative is a Winsorized mean. The highest and lowest extremes are
replaced by the next-to-highest value and by the next-to-lowest value.
For Univariate outliers, Tabachnick and Fidell (2001) suggest changing the scores on the variables for the outlying cases so they are deviant. E.g. if the two largest scores in the distribution are 125 and 122 and the next largest score 87. recode 122 as 88 and 125 as 89.
Outliers
Steps on SPSS 1. Analyze 2. Descriptive3. Explore 4. Statistics ……plots 5. Outliers
Missing Data Any systematic event external to the respondent (such as
data entry errors or data collection problems) or action on the part of the respondent (such as refusal to answer) that leads to missing data.
It means that analyses are based on fewer study participants than were in the full study sample. This, in turn, means less statistical power, which can undermine statistical conclusion validity-the degree to which the statistical results are accurate.
Missing data can also affect internal validity-the degree to which inferences about the causal effect of the dependent variable on the dependent variable are warranted, and also affect the external validity-generalizability.
Strategies to avoid Missing Data Persistent follow-up Flexibility in scheduling appointments Paying incentives. Using well-proven methods to track people
who have moved. Performing a thorough review of
completed data forms prior to excusing participants.
Factors to consider in designing a missing values strategy Extent of missing data Pattern of missing data Nature of missing data. Role of the variable Level of measurement of the variable.
Extent of missing data Researchers usually handle the problem
differently if there is only 1% missing data as opposed to , say, 25% missing.
Pattern of missing data It is more straightforward to deal with data
that are missing a haphazard, random fashion, as opposed to a systematic fashion that typically reflects a bias.
Different patterns of missing data: Missing completely at random (MCAR) Missing at random (MAR) Missing not at random (MNSR).
Missing Completely at Random (MCAR) It means that the probability that the observation
is missing is completely unrelated to either the value of the missing case or the value of any other variables.
Occurs when cases with missing values are just a random subsample of all cases in the sample.
When data are MCAR, analyses remain unbiased, although power is reduced.
E.g. When one participant did not show up post the intervention due to emergency. In this situation, the missing values are not related to the main variable or to the value of other characteristics, such as the person’ s age, sex or experimental group status.
Missing at Random (MAR)
It considered MAR if missingness is related to other variables-but not related to the value of the variable that has the missing values.
This pattern is perhaps the most prevalent pattern of missingness in clinical research.
E.g. men were less likely to keep their follow-up appointment. Thus, missingness is related to a person’s gender.
Missing not at Random (MNSR) A pattern in which the value of the
variable that is missing is related to its missingness. This is often found for such variables as income (not to tell the truth).
Nature of Missing Data.
For only one item is a multi-item-measure. Sometimes an entire variable is missing. In other situations, all data are missing for
study participants.
Role of the variable How one handles the missing data
problem may depend on whether a variable is considered a primary outcome, a secondary outcome, an independent (predictor) variable, or control variable (covariate).
Level of Measurement of the Variable Some strategies are best applied when the
variable is measured on an interval or ratio scale, while others only make sense for nominal-level variables.
Techniques for Handling Missing Data Deletion techniques. Involve excluding subjects
with missing data from statistical calculation. Imputation techniques. Involve calculating an
estimate of each missing value and replacing, or imputing, each value by its respective estimate.
Note: techniques for handling missing data often vary in the degree to which they affect the amount of dispersion around true scores, and the degree of bias in the final results. Therefore, the selection of a data handling technique should be carefully considered.
Deletion Techniques Deletion methods involve removal of cases or variables
with missing data. Listwise deletion. Also called complete case analysis. It is
simply the analysis of those cases for which there are no missing data. It eliminates an entire case when any of its items/variables has a missing data point, whether or not that data point is part of the analysis. It is the default of the SPSS.
Pairwise deletion. Called the available case analysis (unwise deletion). Involves omitting cases from the analysis on a variable-by-variable basis. It eliminates a case only when that case has missing data for variables or items under analysis.
Note: deletion techniques are widely criticized because they assume that the data are MCAR (which is very difficult to ascertain), pose a risk for bias, and lead to reduction of sample size and power.
Imputation Techniques Imputation is the process of estimating
missing data based on valid values of other variables or cases in the sample.
The goal of imputation is to use known relationship that can be identified in the valid values of the sample to help estimate the missing data
Types of Imputation Techniques Using prior knowledge. Inserting mean values. Using regression Expectation maximization (EM). Multiple imputation.
Prior Knowledge Involves replacing a missing value with a
value based on an educational guess. It is a reasonable method if the researcher
has a good working knowledge of the research domain, the sample is large, and the number of missing values is small.
Mean Replacement Also called median replacement for
skewed distribution. Involves calculating mean values from a
available data on that variable and using them to replace missing values before analysis.
It is a conservative procedure because the distribution mean as a whole does not change and the researcher does not have to guess at missing values.
Mean Replacement Advantages:
Easily implemented and provides all cases with complete data.
A compromise procedure is to insert a group mean for the missing values.
Disadvantages: It invalidates the variance estimates derived from the
standard variance formulas by understanding the data’s true variance.
It distorts the actual distribution of values. It depresses the observed correlation that this variable
will have with other variables because all missing data have a single constant value, thus reducing the variance.
Using Regression Involves using other variables in the dataset as
independent variables to develop a regression equation for the variable with missing data serving as the dependent variable.
Cases with complete data are used to generate the regression equation.
The equation is then used to predict missing values for incomplete cases.
More regressions are computed, using the predicted values from the previous regression to develop the next equation, until the predicted values from one step to the next are comparable.
Prediction from the last regression are the ones used to replace missing values.
Using Regression Advantages:
It is more objective than the researcher’s guess but not as blind as simply using the overall mean.
Disadvantages: It reinforces the relationships already in the data,
resulting in less generalizability. The variance of the distribution is reduced because the
estimate is probably too close to the mean. It assumes that the variable with missing data is
correlated substantially with missing data is correlated substantially with the other variables in the dataset.
The regression procedure is not constrained in the estimates it makes.
Expectation Maximization For randomly missing data. It is an iterative process that proceeds in two
discrete steps: In the expectation (E) step, the conditional
expected value of the complete data is computed and then given the observed values, such as correlations.
In the Maximization (M) step, these expected values are then substituted for the missing and maximum likelihood estimation is then computed as though there were no missing data.
Multiple Imputation It produces several datasets and analyzes
them separately. One set of parameters is then formed by
averaging the resulting estimates and standard errors.
Multiple Imputation Advantages:
It makes no assumptions about whether data are randomly missing but incorporates random error because it requires random variation in the imputation process.
It permits use of complete-data methods for data analysis and also includes the data collector’s knowledge.
It permits estimates of nonlinear models. It simulates proper inference from data and increases
efficiency of the estimates by minimizing standard errors.
It is the method of choice for databases that are made available for analyses outside the agency that collected the data.
Multiple Imputation Disadvantages:
It requires conceptual intensiveness to carry out MI, including special software and model building.
It does not produce a unique answer because randomness is preserved in the MI process, making reproducibility of exact results problematic.
it requires large a mounts of data storage space that often exceeds space on personal computers’ hard driver.
