Figure 4.6 (page 119)Typical ways of presenting frequency graphs and descriptive statistics

Sample variance as an unbiased statistic

• Sample variance s2 is an unbiased estimator for the population variance σ2 of the underlying

• Average value of the sample variance will equal the population value

• Sample standard deviation is biased but is still a good estimate of population

Table 4.1 (p. 120) The set of all the possible samples for n = 2 selected from the population described in Example 4.7. The mean is computed for each sample, and the variance is computed two different ways: (1) dividing by n, which is incorrect and produces a biased statistic; and (2) dividing by n – 1, which is correct and produces an unbiased

statistic. From a population of scores 0,0,3,3,9,9 with = 4 and2 = 14.

36/9 = 4 63/9 = 7.0 126/9 = 14

Factors that Affect Variability

1. Extreme scores

2. Sample size

3. Stability under sampling

4. Open-ended distributions

Mean and Standard Deviation as Descriptive Statistics

• If you are given numerical values for the mean and the standard deviation, you should be able to construct a visual image (or a sketch) of the distribution of scores.

• As a general rule, about 70% of the scores will be within one standard deviation of the mean, and about 95% of the scores will be within a distance of two standard deviations of the mean.

• It is common to talk about descriptive statistics as the mean plus or minus the standard deviation.

• For example 36 + or 4 as shown in figure 4.7.

Figure 4.7 (page 122) Caution the data distribution needs to be symmetrical for this to work.Imagine what this looks like with extremely skewed data.

Normal Curve with Standard Deviation | + or - one s.d. |

Properties of the Standard Deviation– If a constant is added to every score in a distribution, the standard

deviation will not be changed. – If you visualize the scores in a frequency distribution histogram,

then adding a constant will move each score so that the entire distribution is shifted to a new location.

– The center of the distribution (the mean) changes, but the standard deviation remains the same.

1 2 3 4 6 7 8 9

Add 5 points to each score

M = 2.5 s = 1.05

M = 7.5 s = 1.05

Properties of the Standard Deviation• If each score is multiplied by a constant, the standard deviation will be

multiplied by the same constant. • Multiplying by a constant will multiply the distance between scores, and

because the standard deviation is a measure of distance, it will also be multiplied.

1 2 3 4

M = 2.5 s = 1.05

Multiply each score by 5

5 10 15 20

M = 12.5 s = 5.24

Properties of the Standard Deviation

• Transformation of scale– Can change the scale of a set of score by adding a constant

• Exam with a mean of 87

• Add 13 to every score

• Exam mean changes to 100

– Can change the scale by multiplying by a constant• Exam with standard deviation of 5

• Multiply each score by 2

• Exam mean changes to 10

Table 4.2 (p. 124) The number of aggressive responses of male and female children after viewing cartoons. Example of APA tale format when reporting descriptive data. Did type of cartoon make a difference on aggressive responses? How do males and females differ?

Figure 4-8 (p. 125) Using variance in inferential statisticsGraphs showing the results from two experiments. In Experiment A, the variability within samples is small and it is easy to see the 5-point mean difference between the two samples. In Experiment B, however, the 5-point mean difference between samples is obscured by the large variability within samples.

Chapter 5: z-scores – Location of Scores and Standardized Distributions

2014 Boston Marathon, Wheelchair Race

• “Ernst Van Dyk (RSA) and Tatyana McFadden (USA) won the men’s and women’s titles at the 118th B.A.A. Boston Marathon. Capturing an unprecedented tenth Boston Marathon title, Van Dyk led from wire-to-wire. McFadden celebrated her 25th birthday by defending her title in the women’s race, just eight days after winning and breaking her own course record at the London Marathon.”

• http://www.baa.org/news-and-press/news-listing/2014/april/2014-boston-marathon-push-rim-wheelchair-race.aspx

• Using SPSS to get standardized scores

z-Scores and Location

• By itself, a raw score or X value provides very little information about how that particular score compares with other values in the distribution.

• To interrupt a score of X = 76 on an exam need to know– mean for the distribution – standard deviation for the distribution

• Transform the raw score into a z-score, – value of the z-score tells exactly where the score is located– process of changing an X value into a z-score– sign of the z-score (+ or –) identifies whether the X value is

located above the mean (positive) or below the mean (negative)

– numerical value of the z-score corresponds to the number of standard deviations between X and the mean of the distribution.

Two distributions of exam scores. For both distributions, µ = 70, but for one distribution, σ = 12. The position of X = 76 is very different for these two distributions.

Figure 5.2 page 140With a box for individual scores. Two distributions of exam scores. For both distributions, µ = 70, but for one distribution, σ = 12. The position of X = 76 is very different for these two distributions.

A

B

z-Scores and Location

• The process of changing an X value into a z-score involves creating a signed number, called a z-score, such that:– The sign of the z-score (+ or –) identifies whether the X

value is located above the mean (positive) or below the mean (negative).

– The numerical value of the z-score corresponds to the number of standard deviations between X and the mean of the

distribution.

Z-scores and Locations

• Visualize z-scores as locations in a distribution.– z = 0 is in the center (at the mean)– extreme tails

• –2.00 on the left

• +2.00 on the right

• most of the distribution is contained between z = –2.00 and z = +2.00

• z-scores identify exact locations within a distribution• z-scores can be used as

– descriptive statistics – inferential statistics

Figure 5-3 (p. 141)The relationship between z-score values and locations in a population

distribution.

One S.D. Two S.D.