Range
Range=XH−XL (Formula 4.1) Interquartile range, IQR IQR=Q3−Q1 (Formula 4.2) Semi-interquartile range, SIQR SIQR=Q3−Q1
2 (Formula 4.3) Mean deviation for population, MD MD=Σ X−μ
N (Formula 4.4) Mean deviation for sample, MD MD=Σ X−M
n (Formula 4.5)
118 4 Measures of Variability
The definitional formulas for the variance Population variance,σ2
σ2=Σ X−μ2
N (Formula 4.6) Sample variance
s2=Σ X−M 2
n−1 (Formula 4.7)
The deviation score formulas for the variance Population variance Sample variance σ2=Σx2
N (Formula 4.8) s2= Σx2
n−1 (Formula 4.9) The sum of squares formulas for the variance Population variance Sample variance σ2=SS
N (Formula 4.10) s2= SS
n−1 (Formula 4.11) The computational formulas for the variance Population variance
σ2=ΣX2− ΣX 2 N
N (Formula 4.12)
Sample variance s2=ΣX2− ΣX 2 n
n−1 (Formula 4.13)
Any formula for the standard deviation is the square root of the variance formula.
Key Terms
Measures of variability (or dispersion)
Variance Range
Definitional formulas Quartile
Sum of squares (SS) Percentile
Computational (raw score) formulas
Interquartile range Standard deviation Semi-interquartile range 68-95-99.7 rule Mean deviation
Questions and Exercises
1 Given two samples, one in whichn= 36, the other wheren= 60, which dis- tribution would have the larger variance? Which variance is likely to be closer in value to the population variance? Which sample is likely to have the larger range?
2 Assume two samples:M= 78 andM= 155. Which sample would have the larger variance?
3 A school psychologist wants to inform a teacher about the mean and stand- ard deviation of the students’IQ scores. The scores are below; assume they are a sample.
IQ scores 98, 111, 102, 100, 101, 109 a What is the mean?
b What is the standard deviation?
4 Calculate the range, variance, and standard deviation of this sample of scores.
2, 4, 7, 4, 8, 5, 1, 4, 4, 5 a What is the range?
b What is the variance?
c What is the standard deviation?
5 A researcher who uses heart rate as the dependent variable finds the 75th percentile to be a heart rate of 111 and the 25th percentile to be at 81.
a Compute theIQR.
b Compute theSIQR.
6 For the following populations of scores, find the mean deviations:
a 5, 7, 9, 9, 13, 14, 15, 16
b 23, 25, 31, 34, 36, 39, 44, 56, 63, 69 c 6, 8, 3, 9, 1, 4, 7, 4, 1, 1, 8, 2
7 Which of the following variance definitions is correct?
a The average of the deviations scored squared.
b The average of the absolute value of the deviations.
c The average of the deviation scores square rooted.
d The average of the squared deviation scores.
e The average deviation score.
120 4 Measures of Variability
8 What does a distribution with a mean of 50 and standard deviation of zero look like?
9 For each situation, specify whether we should usesorσ.
a A set of coaches are interested in the variability of their basketball team’s scores over the season.
b A clinician is evaluating a new treatment for sexual dysfunctions.
c A teacher is interested in providing feedback to students about class per- formance on the midterm exam.
d A manufacturer takes a sample of light bulbs to estimate the variability of their life.
10 Why is the formula for a sample variance different from the formula for a population variance?
11 Calculate the variance and standard deviation for this population of scores.
22, 32, 21, 20, 19, 15, 23
12 Which distribution of sample scores has the larger variance?
DistributionA: 2, 4, 5, 1, 1, 2, 3, 9 DistributionB: 34, 39, 34, 35, 33, 32
13 A negatively skewed distribution has a mean of 500 and a standard devi- ation of 100. Given what we have learned in this chapter, is it possible to determine the percentage of scores that fall between 400 and 600? If so, what is it?
14 What is the main disadvantage in using the range as a measure of dispersion?
15 As a descriptive statistic, is the variance or the standard deviation a better measure of variability? Why?
16 What is the standard deviation of this population of scores?
9, 7, 10, 14, 12, 9, 16, 13, 11
17 If a normal distribution has a mean of 50 and a standard deviation of 10, what scores encapsulate the middle 68% of the distribution? The middle 95% of the distribution? The middle 99.7% of the distribution?
18 What if a normal distribution has the same mean as in question 17, but had a standard deviation of 2. What scores would encapsulate the middle 68, 95, and 99.7% of scores?
19 If a normal distribution has a mean of 80 and 68% of the scores are between 68 and 92, what is the variance of that distribution?
20 If a normal distribution has a variance of 100 and 95% of the scores are between the values of 120 and 160, what is the mean?
21 For a set of 10 000 scores that is normally distributed and has aμof 100 and aσof 15, about how many of the scores will be:
a Greater than 130?
b Greater than 115?
c Greater than ±3σaway from 100?
d Greater than ±2σaway from 100?
22 If a distribution has aM= 4.5 ands2= 1.6, what would be theMands2if all the raw scores have 10 added to them?
23 Refer to the data found in Chapter 3, Box 3.2. Compute the standard devia- tions of the experimental and control groups, for each phase of the study.
a Baseline b Post-testing
24 An experiment is conducted to evaluate the effectiveness of two different atti- tude change techniques. The dependent variable is attitudes toward immi- grants. In the following table, higher numbers reflect more positive attitudes.
TechniqueA TechniqueB
Pretest Posttest Pretest Posttest
3 7 2 4
4 4 3 2
5 6 4 5
2 5 3 3
TechniqueA Calculate:
a PretestM b Pretests2 c Pretests
122 4 Measures of Variability
d PosttestM e Posttests2 f Posttests TechniqueB Calculate:
a PretestM b Pretests2 c Pretests d PosttestM e Posttests2 f Posttests
25 Complete the following table.μ= 50 andσ= 5. The constants specified are used to transform the scores of the distribution.
X +10 X–10 X 10 X ÷10
μ= ? μ= ? μ= ? μ= ?
σ2= ? σ2= ? σ2= ? σ2= ?
26 What if a newcomer to American football decided to record the yardage gained or lost on each play of a football game in terms of feet instead of the more typical measure of yards; what would the coach need to do with the data to compare the team’s performance with previous games?
27 Three friends sampled students at their university to see how much vari- ability there is in daily time spent on social media. One asked 25 people and got anσof 10 minutes; another asked 50 people and got anσof 15 minutes;
and a third asked 500 people and got an σ of 25 minutes. All three attempted to randomly sample the student body. What is our best guess of the actual population standard deviation?
Computer Work
Determine the range, interquartile range, and semi-interquartile range for each of the following sample distributions.
28 Scores:
13 5 11 17 8 10 13 12 15
15 18 19 16 14 11 12 11 10
(Continued)
14 4 5 15 11 10 19 13 14
7 8 5 11 11 9 9 14 15
8 11 17 17 10 9 8 16 14
7 18 6 17 18 18 11 6 13
29 Scores:
43 45 51 27 48 27 43 22 25
45 38 19 26 24 56 42 53 47
54 48 25 39 51 30 29 33 39
27 58 35 33 21 39 35 34 35
18 19 57 51 40 29 28 46 26
37 55 26 47 35 46 53 36 23
Determine the mean, variance, standard deviation, and range for each of the fol- lowing sample distributions.
30 Scores:
3 5 3 7 9 10 2 12 15
1 8 9 6 4 11 1 11 10
1 4 5 5 3 10 9 13 14
31 Scores:
102 100 99 81 75 113 100
106 114 82 79 88 111 104
100 106 85 99 82 101 100
32 Scores: (For this question only, let us diverge from our commitment to only go out two decimal places–instead, let us go out four, since the values are so small.)
0.1070 0.2190 0.1917 0.2120 0.2016 0.1432 0.1939 0.0988 0.2002 0.1859 0.0847 0.1965 0.1492 0.1861 0.0854 0.1656 0.1776 0.1517 0.1942 0.1812 0.1911 (Continued)
124 4 Measures of Variability
33 Scores:
1020 1000 990 810 750 1130 1000
1060 1140 820 790 880 1110 1040
1000 1060 850 990 820 1010 1000
34 Multiply each value in the data set for Work Problem #32 by three. Find the new mean, variance, standard deviation, and range.
35 Divide each value in the data set for Work Problem #32 by three. Find the new mean, variance, standard deviation, and range.