Measures of Variability

OVERVIEW

Measures of variability describe the average dispersion of data around a mean

most common = range, standard deviation and the standard error of the mean

Summary

RANGE

PERCENTILES

STANDARD DEVIATION (SD)

a measure of the average spread of individual values around the sample or population mean
calculated by squaring the differences between each value and the sample mean, summing them then dividing the result by n – 1 to give the variance
SD = the square root of the variance
SD important because:

reporting the SD along with the mean, gives an indication at a glance as to whether the sample mean represents a real trend in the sample
if the sample is randomly selected and large -> it can be assumed to be close to that of the population
the SD is used to calculate the standard error (see below)
any data point from a normal distribution can be described as a multiple of standard deviations from the population mean

tables will then tell us the proportion of the distribution with values more extreme than that (z transformation)

STANDARD ERROR

Standard error is an estimate of the spread of sample means around the population mean
it is estimated from the data in a single sample
it is an estimated prediction based on the number in the sample and the sample sd

SE = SD / square root of n

thus, the variability among sample means will be increased if there is
- (a) a wide variability of individual data and
- (b) small samples
SE used in parametric tests to quantify the difference between a sample mean & its proposed population mean, i.e. how far the two are apart in multiples of the SE (z-transformation)
SE is used to calculate confidence intervals

CONFIDENCE INTERVALS

CI is the range around a sample mean within which you predict the means of the sample’s population lies
the range in which you predict the ‘true’ value lies

Calculation

95% of sample means should lie between 1.96 standard error of the mean above & below their sample mean
thus, if the sample is large enough and is normally distributed as long as the sample was randomly selected then it should also represent the 95% CI for the population mean
the population mean doesn’t fall within this range -> there is a 95% chance that the samPle is from a different population

Information provided

an indication of the precision of the sample mean as an estimate of the population mean
the wider the CI, the greater the imprecision, the greater the potential difference between the calculated sample mean & ‘true’ mean

Causes of wide CI’s