Volume I - Section VI - Statistical Analysis 
Page VI - 3 
Distribution of Sample Means 
Thus, a standard deviation of sample means describes the variability, or spread, of sample means 
about the true population mean. In most practical situations, however, only one sample mean is 
available. 
The standard error of the mean is calculated by dividing the standard deviation by the square root 
of the number of observations. 
standard error of means = standard deviation/ square root(n) 
The resulting estimate of the standard deviation of sample means is called the standard error of 
means and can be interpreted in a manner similar to the standard deviation of raw scores. For 
example, the probability of obtaining a sample z-score mean that is outside the -1.96 to +1.96 
range is 5 out of 100. 
The concept of statistical significance is based on the assumption that events that occur very 
infrequently by chance are “significant.” For example, Tchebysheff s Theorem states that ‘given 
a number k greater than or equal to 1 and a set of n measurements x\, X 2 , at least [ l-( 1/A: 2 )] of 
the measurements will lie within k standard deviations of their mean’. For example, we expect 
three out of four values to lie within two standard deviations of the mean and eight out of nine 
within three standard deviations of the mean. 
6.1.6 Standard Error of the Difference 
Another situation involves the comparison of two means. Consider a situation where samples are 
repeatedly drawn from two groups. For example, repeatedly testing male and female groups on a 
mathematics test, resulting in 100 sample means for male groups and 100 sample means for 
female groups. Subtracting the male mean from the female mean for 100 pairs of groups will 
result in 100 differences. A standard deviation of these differences could be computed and 
interpreted in a manner similar to the standard deviation of scores and to the standard error of the 
mean. 
