Psychology2016

A-8 APPENDIX A

when there are extreme scores in the distribution. For example, if you look at Table A.2,

the range of those IQ scores would be 160 – 95, or 65. But if you just look at the numbers,

you can see that there really isn’t that much variation except for the three highest scores

of 139, 150, and 160.

The other measure of variability that is commonly used is the one that is related to

the normal curve, the standard deviation. This measurement is simply the square root

of the average squared difference, or deviation, of the scores from the mean of the distri-

bution. The mathematical formula for finding the standard deviation looks complicated,

but it is really nothing more than taking each individual score, subtracting the mean from

it, squaring that number (because some numbers will be negative and squaring them gets

rid of the negative value), and adding up all of those squares. Then this total is divided

by the number of scores, and the square root of that number is the standard deviation. In

the IQ example, it would go like this:

Standard Deviation Formula SD = "[S(X − M)^2 #N]

The mean (M) of the 10 IQ scores is 114.6. To calculate the standard deviation, we

1. Subtract the mean from each score to get a deviation score S (X − M)


2. Square each deviation score S (X − M)^2


3. Add them up. Remember that’s what the sigma (S) indicates S S(X − M)^2


4. Divide the sum of the squared deviation by N (the number of scores) S S(X − M)^2 #N


5. Take the square root ("X) of the sum for our final step. "[S(X − M)^2 #N]


6. The process is laid out in Table A.3.
   The standard deviation is equal to 23.5. What that tells you is that this particular
   group of data deviates, or varies, from the central tendencies quite a bit—there are some
   very different scores in the data set or, in this particular instance, three noticeably differ-
   ent scores.

standard deviation

the square root of the average squared

deviations from the mean of scores in

a distribution; a measure of variability.

Table A.3 Finding the Standard Deviation

Score Deviation from The Mean (X 6 M) Squared Deviation

160.00 45.40 2,061.16

(ex. 160 − 114.60 = 45.40) (45.40^2 = 2,061.16)

150.00 35.4 1,253.16

139.00 24.4 595.36

102.00 −12.60 158.76

102.00 −12.60 158.76

100.00 −14.60 213.16

100.00 −14.60 213.16

100.00 −14.60 213.16

98.00 −16.60 275.56

95.00 −19.60 384.16

Sum of Scores

(SX) = 1,146.00

Mean = (SX)#N

= 1,146# 10 = 114.60

S(X − M) = 0.00 S(X − M) (^2) = 5,526.40
Standard Deviation
= " [S(X^ −^ M)^2 #N]
= "5,526.40#^10 = 23.5
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