Sample size affects the SD of the DOSM

The standard deviation of the DOSM falls as: 

1/ (sqroot of n)

The DOSSD

● Note: the DOSSD is not normally distributed!

● The sample variance and SD are always positive,

so the DOSSD cannot contain any numbers < 0.

The DOSSD for large sample sizes

● When the sample size is large, the mean of the

DOSSD is about equal to the population SD and the

variance of the DOSSD is small.

The DOSSD for small sample sizes

● When the sample size is small, the mean of the

DOSSD is smaller than the population SD and the

variance of the DOSSD is large.

Important facts about the DOSM & DOSSD

● The mean of the DOSM is always equal to the

population mean.

● The SD of the DOSM depends on the size of the

sample, n.

● The mean of the DOSSD underestimates the

population SD, especially for small n.

● An “adjustment factor” is used to adjust the

sample SD so that it is closer to the population

SD.

● This is the n-1 term we saw earlier in the

denominator of the standard deviation.

● The variance of the DOSSD depends on n

Why we care about the SD of the DOSM

● The DOSM tells us how the means of many samples

(of size n) will be distributed, relative to the

population mean.

● The mean of the DOSM is always equal to the

population mean

● The SD of the DOSM depends on:

● The SD of the population.

● The sample size (n).

● For a given population SD, the SD of the

DOSM will be inversely proportional to the

square root of n.

The standard error

● The standard deviation of the distribution of

sample means tells us how close the sample mean is likely to be from the population mean.

● Because this is an important diagnostic tool, the SD of DOSM: the Standard Error.

The variance of the DOSSD

● Recall that the distribution of sample standard

deviations is not distributed normally. so the

variability of standard deviations across samples is

given by the variance of the DOSSD.

● The variance of the DOSSD tells us how the

standard deviations of many samples will be

distributed relative to the population SD.

● The variance of the DOSSD does not have a special

name, because no one calculates it explicitly.

The mean is an unbiased statistic

● The variability of the means and standard

deviations obtained across different samples is

determined by the sample size, n.

● In general, as n gets larger the variability of the

descriptive statistics gets smaller.

● The mean is an unbiased statistic.

● The variability of the sample mean (across

samples) depends on n, but the mean of any given

sample is equally likely to be above or below the

population mean.

The variance is a biased statistic

● The variance and standard deviations are biased

statistics.

● The sample variance systematically underestimates

the population variance.

● Therefore, the sample standard deviation is

likely to be less than the population standard

deviation.

● This effect becomes more pronounced as the

sample size gets smaller.