chapter 5: variability and standard (z) scores how do we quantify the variability of the scores in a...
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Chapter 5: Variability and Standard (z) Scores
How do we quantify the variability of the scores in a sample?
55 60 65 70 75 80 85 90 95 100 105 110 1150
1
2
3
4
5
Ice Dancing Score
Fre
quen
cy
Method 1: range: difference between the highest and lowest scores
Ice Dancing , compulsory dance scores, Winter Olympics
111.15108.55
106.6103.33100.06
97.3896.6796.1292.7589.6285.3684.5883.8983.1280.47
80.379.3176.7374.2572.0168.8763.7359.64
Example: The range of ice dancing scores is 111.15-59.64 = 51.51 points
The range is easy to calculate, but it really only depends on two scores. So it’s not a very informative or reliable measure of variability.
Method 2: The semi-interquartile range (Q): One half of the distance between P75 and P25.
Ice Dancing , compulsory dance scores, Winter Olympics
Example: ice dancing scores
Q1 = P25
Q3 = P75
213 QQ
Q
Percentile rank111.15 98108.55 93
106.6 89103.33 85100.06 80
97.38 7696.67 7296.12 6792.75 6389.62 5985.36 5484.58 5083.89 4683.12 4180.47 37
80.3 3379.31 2876.73 2474.25 2072.01 1568.87 1163.73 759.64 2
Score
pLpH
PLpSLSHSL )(
38.772428
2425)73.7631.79(73.76251
PQ
20.977276
7275)67.9638.97(67.96753
PQ
91.92
38.7720.97
213
QQQ
Method 3: Variance: the mean of the squares of the deviation scores
deviation score: The difference between a score and the mean of the scores )( XX
Formula for variance of a population of scores
Formula for variance of a sample of scores
Sums of squared deviation scores 2)( XXSSX
N
SS
N
XXX
X
22 )(
n
SS
n
XXs XX
2
2 )(
Example: find the variance of this sample of 7 numbers: 5,3,1,6,2,8,3
4X
n
SS
n
XXs XX
2
2 )(
14.57
36
7
116449117
)1()4()2()2()3()1()1(
7
)43()48()42()46()41()43()45()(
2222222
222222222
n
XXSX
Calculating variance this way can be tedious. Fortunately there’s a shortcut for calculating SSx:
Sum of squared deviations from the mean
Sum of squares
Sum squared divided by n
n
XXXXSSX
2
22)(
Example: from this sample of 7 numbers: 5,3,1,6,2,8,3
4X
n
XXXXSSX
2
22)(
3611644911
)1()4()2()2()3()1()1(
)43()48()42()46()41()43()45()(2222222
22222222
XX
78428)326135( 222 X
36112148
7
784148
2
2 n
XX
1483826135 22222222 X
Example: calculate the variance of this sample of 10 numbers:
8 6 3 7 1 7 7 8 9 10
X X2
2X
X
n
SSS XX2
2X
8 646 363 97 491 17 497 498 649 81
10 100
10
502
66
4356
435.6
66.4
6.64
n =
n
X2
n
XXXXSSX
2
22)(
n
XXSSX
2
2
standard deviation: the square root of the variance
Formula for standard deviation for a population of scores
Formula for standard deviation for a sample of scores
The standard deviation has the same units as the original scores (e.g. points, inches, etc.)
N
SS
N
XXX
X
2)(
n
SS
n
XXS XX
2)(
Warning! Point of future confusion!
The definition of variance and standard deviation has an (or N) in the denominator.
Later when we get in to inferential statistics, we’ll start dividing by n-1:
The first definition is the true average of the squared deviance from the mean. But this number a biased estimate of the variance of the population.
Divide by ‘n’ when you just want the standard deviation of our sample (or population). Divide by ‘n-1’ when you want to estimate the standard deviation of the population.
n
SS
n
XXS XX
2)(
11
)( 2
n
SS
n
XXs XX
N
SS
N
XXX
X
2)(
Example: calculate the standard deviation of this sample of 10 numbers:
8 6 3 7 1 7 7 8 9 10
X X2
2X
X
n
SSS XX2
2X
8 646 363 97 491 17 497 498 649 81
10 100
10
502
66
4356
435.6
66.4
6.64
2.58
n =
n
X2
2XX SS
n
XXSSX
2
2
Example: calculate the standard deviation of this sample of 20 numbers:
863717789
103234632831
6436
949
149496481
100949
1636
94
6491
X X2
2X
X
n
SSS XX2
2X
n =
n
X2
2XX SS
20
663
101
10201
510.05
152.95
7.65
2.77
n
XXSSX
2
2
Characteristic RangeSemi-
interquartile range
Standard deviation
Frequency of use Some Very little Almost always
Mathematical tractability
Very little Very little Great
Sampling stability Worst OK Best
Use with skewed distributions
Not so good OK Interpret with caution
Most closely related central
tendency
None Median Mean
Use with open ended
distributions
No OK No
Affected by sample size
Yes No No
Ease of calculation Easy OK OK
Fun facts about the standard deviation:
Adding a constant to each number in a sample does not change the standard deviation (or variance)
SX+b = SX
Multiplying each number in a sample by a constant multiplies the standard deviation by that same constant.
SaX = aSX
How big is a standard deviation?
For a normal (bell-shaped) distribution:
68.2% of the values fall within one standard deviation of the mean95.4% of the values fall within two standard deviations of the mean99.7% of the values fall within three standard deviations of the mean
1 standard deviation above and below the mean is where the bend of the curve switches (the ‘inflection point’)
80 90 100 110 1200
20
40
60
80
100
120
140
160
Score
Guess the mean and standard deviation
80 90 100 110 1200
20
40
60
80
100
120
140
160
Score
Mean= 99, s.d. = 8.0
Guess the mean and standard deviation
0 50 1000
10
20
30
40
50
60
Score
Guess the mean and standard deviation
0 50 1000
10
20
30
40
50
60
Score
Mean= 60, s.d. = 27.3
Guess the mean and standard deviation
-400 -300 -200 -100 0 100 2000
50
100
150
Score
Guess the mean and standard deviation
-400 -300 -200 -100 0 100 2000
50
100
150
Score
Mean= -99, s.d. = 99.7
Guess the mean and standard deviation
497 498 499 500 501 5020
50
100
150
Score
Guess the mean and standard deviation
497 498 499 500 501 5020
50
100
150
Score
Mean= 500, s.d. = 1.0
Guess the mean and standard deviation
Guess the mean and standard deviation
-4 -2 0 2 4 60
20
40
60
80
100
Score
Guess the mean and standard deviation
-4 -2 0 2 4 60
20
40
60
80
100
Score
Mean= 1, s.d. = 1.9
Standard Scores (z scores)
Sometimes it is useful to compare scores across distributions that have different means and standard deviations. A common way to do this is to convert the scores into standard deviation units, or ‘z scores’.
The goal is to modify all of the scores so that the new mean is equal to zero, and the new standard deviation equal to one.
To make the new mean zero, we subtract the mean from all scores. Remember this shifts the mean but doesn’t change the standard deviation.
To make the new standard deviation equal to 1, we divide all scores by the standard deviation. This would normally change the mean, but since it’s zero, it doesn’t change.
Here’s the formula for changing a sample of scores, X to z:
XS
XXz
Example: Convert the following ten scores to z scores
X X2
2X
X
n
SSS XX2
2X
n =
n
X2
2XX SS
Step 1, calculate the mean and standard deviation:
234
12426293
723
854
52916
144176438448649
49529
642916
10
18504
328
107584
10758.40
7745.60
774.56
27.83
32.80n
XX
n
XXSSX
2
2
Example: Convert the following ten scores to z scores
X 2XX SS
Step 2, for each score, subtract the mean and divide by the standard deviation
234
12426293
723
854
27.83
32.80n
XX
-0.35-1.03-0.750.331.052.16
-0.93-0.35-0.890.76
-9.80-28.80-20.80
9.2029.2060.20
-25.80-9.80
-24.8021.20
XX XS
XXz
Check for yourself that the mean of z is 0, and the standard deviation is 1.
Z-transforming your scores doesn’t affect the shape of the distribution.
0 50 100 1500
20
40
60
80
100
Score
Mean= 80, s.d. = 33.0
-2 -1 0 1 2 30
20
40
60
80
100
z score
Mean= 0, s.d. = 1
The standard normal distribution
The standard normal distribution is a continuous distribution.It has a mean of 0 and a standard deviation of 1The total area under the curve is equal to 1
-4 -3 -2 -1 0 1 2 3 4
Rel
ativ
e fr
equ
enc
y
z score
-3 -2 -1 0 1 2 3
area =0.1587
z
-3 -2 -1 0 1 2 3
area =0.3413
z
Table A (page 436) gives you the proportion of scores for given ranges in the standard normal
Column 2 Area between 0 and z
Column 3Area above z