CHAPTER 1 Overview and Descriptive Statistics

the divisor is used rather than n. In other words, if we used a divisor n in the sample variance, then the resulting quantity would tend to underestimate (produce estimated values that are too small on the average), whereas dividing by the slightly smaller corrects this underestimating.

It is customary to refer to s2 as being based on degrees of freedom (df). This terminology reflects the fact that although s2 is based on the n quantities

, these sum to 0, so specifying the values of any of the quantities determines the remaining value. For example, if and

, , and , then automatically , so only three of the four values of are freely determined (3 df).

A Computing Formula for s2

It is best to obtain s2 from statistical software or else use a calculator that allows you to enter data into memory and then view s2 with a single keystroke. If your calcula- tor does not have this capability, there is an alternative formula for Sxx that avoids calculating the deviations. The formula involves both , summing and then squaring, and , squaring and then summing.gxi

2 AgxiB2

xi 2 x x3 2 x 5 2×4 2 x 5 24×2 2 x 5 26×1 2 x 5 8

n 5 4 n 2 1×1 2 x, x2 2 x, c, xn 2 x

n 2 1 n 2 1

s2 n 2 1

An alternative expression for the numerator of s2 is

Sxx 5 g (xi 2 x) 2 5 gxi

2 2 AgxiB2

n

Proof Because . Then,

Traumatic knee dislocation often requires surgery to repair ruptured ligaments. One measure of recovery is range of motion (measured as the angle formed when, start- ing with the leg straight, the knee is bent as far as possible). The given data on post- surgical range of motion appeared in the article “Reconstruction of the Anterior and Posterior Cruciate Ligaments After Knee Dislocation” (Amer. J. Sports Med., 1999: 189–197):

The sum of these 13 sample observations is , and the sum of their squares is

Thus the numerator of the sample variance is

from which and . ■

Both the defining formula and the computational formula for s2 can be sensitive to rounding, so as much decimal accuracy as possible should be used in intermediate calculations.

Several other properties of s2 can enhance understanding and facilitate com- putation.

s 5 11.47s2 5 1579.0769/12 5 131.59

Sxx 5 gxi 2 2 [(gxi)

2]/n 5 222,581 2 (1695)2/13 5 1579.0769

gxi 2 5 (154)2 1 (142)2 1 c 1 (122)2 5 222,581

gxi 5 1695

154 142 137 133 122 126 135 135 108 120 127 134 122

5 gxi 2 2 2x # nx 1 n(x)2 5 gxi2 2 n(x)2

g (x i 2 x ) 2 5 g (x i

2 2 2x # x i 1 x 2) 5 gx i2 2 2x gx i 1 g (x)2 x 5 gxi /n, nx

2 5 AgxiB2/n

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Measures of Variability 39 PROPOSITION Let be a sample and c be any nonzero constant. 1. If , then , and 2. If , then where is the sample variance of the x’s and is the sample variance of the y’s.sy 2sx 2 sy 2 5 c2sx
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