Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in...
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Transcript of Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in...
![Page 1: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/1.jpg)
Chapter 4: Variability
![Page 2: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/2.jpg)
Variability
• Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together
![Page 3: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/3.jpg)
Central Tendency and Variability
• Central tendency describes the central point of the distribution, and variability describes how the scores are scattered around that central point.
• Together, central tendency and variability are the two primary values that are used to describe a distribution of scores.
![Page 4: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/4.jpg)
Variability
• Variability serves both as a descriptive measure and as an important component of most inferential statistics.
• As a descriptive statistic, variability measures the degree to which the scores are spread out or clustered together in a distribution.
• In the context of inferential statistics, variability provides a measure of how accurately any individual score or sample represents the entire population.
![Page 5: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/5.jpg)
Variability (cont.)
• When the population variability is small, all of the scores are clustered close together and any individual score or sample will necessarily provide a good representation of the entire set.
• On the other hand, when variability is large and scores are widely spread, it is easy for one or two extreme scores to give a distorted picture of the general population.
![Page 6: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/6.jpg)
![Page 7: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/7.jpg)
Measuring Variability
• Variability can be measured with – the range– the interquartile range– the standard deviation/variance.
• In each case, variability is determined by measuring distance.
![Page 8: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/8.jpg)
The Range
• The range is the total distance covered by the distribution, from the highest score to the lowest score (using the upper and lower real limits of the range).
![Page 9: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/9.jpg)
Range
• URL xmax - LRL xmin
– e.g. 3, 7, 12, 8, 5, 10
![Page 10: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/10.jpg)
Problems?
Distribution 11, 8, 9, 9, 10, 10 R = ?
Distribution 21, 2, 3, 6, 8, 10 R = ?
![Page 11: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/11.jpg)
The Interquartile Range
• The interquartile range is the distance covered by the middle 50% of the distribution (the difference between Q1 and Q3).
![Page 12: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/12.jpg)
Scores
2, 3, 4, 4, 5, 5, 6, 6,
6, 7, 7, 8, 8, 9, 10, 11
![Page 13: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/13.jpg)
x f cf cp c%
11 1 16 16/16 100%
10 1 15 15/16 93.75%
9 1 14 14/16 87.5%
8 2 13 13/16 81.25%
7 2 11 11/16 68.75%
6 3 9 9/16 56.25%
5 2 6 6/16 37.5%
4 2 4 4/16 25%
3 1 2 2/16 12.5%
2 1 1 1/16 6.25%
![Page 14: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/14.jpg)
1
2
3
1 2 3 4 5 6 7 8 9 10 11
![Page 15: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/15.jpg)
Interquartile range3.5 points
Bottom 25%
Top 25%
1
2
3
1 2 3 4 5 6 7 8 9 10 11
Q1 = 4.5 Q3 = 8
![Page 16: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/16.jpg)
The Standard Deviation
• Standard deviation measures the standard (or average) distance between a score and the mean.
![Page 17: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/17.jpg)
0, 1, 3, 8
€
μ = 8 +1+ 3 + 0
4
€
= 3
x8
1
3
0
(x- µ)8 - 3 = +5
1 - 3 = -2
3 - 3 = 0
0 - 3 = -3
5
1
3
0 2 4 6 8
fµ = 3
![Page 18: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/18.jpg)
x
1
0
6
1
x - µ
1 - 2 = -1
0 - 2 = -2
6 - 2 = +4
1 - 2 = -1
(x - µ)2
1
4
16
1
22 = ∑(x - µ)2 = SS
∑x = 8
µ = 2
or
x
1
0
6
1
x2
1
0
36
1
∑x = 8
∑x2 = 38
€
SS = ∑ x 2 − (∑ x)2
N
€
= 38 − 82
4
€
= 38 − 16
€
= 22
![Page 19: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/19.jpg)
15
µ = 6
3
2
1
1 2 3 4 5 6 7 8 9 10X
Fre
quen
cy
![Page 20: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/20.jpg)
• 1, 9, 5, 8, 7
• µ = 6
x
1
9
5
8
7
(x - µ)
1 - 6 = -5
9 - 6 = +3
5 - 6 = -1
8 - 6 = +2
7 - 6 = +1
(x - µ)2
25
9
1
4
1
€
∑(x − μ)2 = 40 = SS
€
σ 2 = SSN
€
= ∑(x − μ )2
N€
= 405
€
= 8
€
σ = SSN
€
= ∑(x − μ )2
N
€
= 2.83
![Page 21: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/21.jpg)
Variance and Standard Deviation for a population of scores
€
= ∑(x − μ )2
N
€
σ = SSN€
σ 2 = SSN
€
= ∑(x − μ )2
N
![Page 22: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/22.jpg)
µ = 40
€
σ =4
![Page 23: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/23.jpg)
x x x x x x x x x
Samplevariability
Sample
Populationvariability
Populationdistribution
![Page 24: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/24.jpg)
1
12
2
3
3
3
44
4
5
5
6
6
7
7 8
8
9
9
Population
€
σ =?
1, 6, 4, 3, 8, 7, 6 Sample
Find the standard deviation ‘s’
![Page 25: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/25.jpg)
Variance and Standard Deviation for a Sample Used to Estimate the
Population ValueVariance:
€
s2 =SSn−1
=(x − x)2∑n−1
€
s =SSn−1
=SSn−1
![Page 26: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/26.jpg)
1, 6, 4, 3, 8, 7, 6,
![Page 27: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/27.jpg)
143
2
1
1 2 3 4 5 6 7 8 9 10X
Fre
quen
cy €
X = 5
![Page 28: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/28.jpg)
1, 6, 4, 3, 8, 7, 6
x
1 1 - 5 = -4 16
6 6 - 5 = +1 1
4 4 - 5 = -1 1
3 3 - 5 = - 2 4
8 8 - 5 = +3 9
7 7 - 5 = +2 4
6 6 - 5 = +1 1
€
(x − X)
€
(x − X)2
€
(x − X)2 = SS = 36∑
€
or SSn−1
€
s tandard deviation s =(x − X)2∑n−1
Sample
€
X =x∑n
=357
= 5
€
var iance s2 =(x − X)2∑n−1
€
or SSn−1
€
= 366
= 6 = 2.45
![Page 29: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/29.jpg)
Sum of Squares
€
s =(x − X)2∑n−1
€
But Also :
SS = x2∑ −( x∑ )2
n
€
s2 =(x − X)2∑n−1€
SS = (x − X)2∑
![Page 30: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/30.jpg)
x x2
1 1
6 36
4 16
3 9
8 64
7 49
6 36
35 211
€
SS = x2∑ −( x∑ )2
n
€
=211−352
7
€
=211−1225
7
€
=211−175
€
=36
![Page 31: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/31.jpg)
€
σ 2 =
€
σ =
€
SSN
=
€
SSN
=
€
(x −μ )2∑N
€
(x −μ )2∑N
€
s2 =
€
s =
€
SSn−1
=
€
SSn−1
=
€
(x − X)2∑n−1
€
(x − X)2∑n−1
![Page 32: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/32.jpg)
Example
• Randomly select a score from a population
x = 47
• What value would you predict for the population mean?
€
if σ = 4
€
if σ = 20
![Page 33: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/33.jpg)
Properties of the Standard Deviation
1. The same score can have very different meanings in 2 different distributions
2. Standard deviation helps us make predictions about sample data
e.g. Figure 4.8low variability
high variability
What is the probability of picking a score near µ = 20 ?
3. Sampling error - how big? (standard deviation a measure)
![Page 34: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/34.jpg)
10 15 20 25 30X
µ = 20
€
σ =2
(a)fr
eque
ncy
Your Score
10 15 20 25 30X
µ = 20
€
σ =6
(b)
freq
uenc
y
Your Score
![Page 35: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/35.jpg)
Transformations of Scale
1. Adding a constant to each score will not change the standard deviation
2. Multiplying each score by a constant causes the standard deviation to be multiplied by the same constant
![Page 36: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/36.jpg)
Comparing Measures of Variability
• Two considerations determine the value of any statistical measurement:
1. The measures should provide a stable and reliable description of the scores. It should not be greatly affected by minor details in the set of data.
2. The measure should have a consistent and predictable relationship with other statistical measurements.
![Page 37: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/37.jpg)
Factors that Affect Variability
1. Extreme scores
2. Sample size
3. Stability under sampling
4. Open-ended distributions
![Page 38: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/38.jpg)
Relationship with Other Statistical Measures
• Variance and standard deviation are mathematically related to the mean. They are computed from the squared deviation scores (squared distance of each score from the mean).
• Median and semi-interquartile range are both based on percentiles and therefore are used together. When the median is used to report central tendency, semi-interquartile range is often used to report variability.
• Range has no direct relationship to any other statistical measure.
![Page 39: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/39.jpg)
Sample variability and degrees of freedom
df = n - 1
![Page 40: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/40.jpg)
The 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.
![Page 41: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/41.jpg)
41
Mean number of errors on easy vs. difficult tasks for males vs. females
Easy Difficult
Female 1.45 8.36
Male 3.83 14.77
![Page 42: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/42.jpg)
When we report descriptive statistics for a sample, we should
report a measure of central tendency and a measure of
variability.
![Page 43: Chapter 4: Variability. Variability Provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.](https://reader036.fdocuments.us/reader036/viewer/2022081506/56649f455503460f94c66f47/html5/thumbnails/43.jpg)
43
Mean number of errors on easy vs. difficult tasks for males vs. females
Easy Difficult
Female M =1.45
SD = .92
M = 8.36
SD = 2.16
Male M =3.83
SD =1.24
M =14.77
SD = 3.45