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
• standard error
• confidence intervals
• z-transformation

RANGE

• smallest & largest values in a sample

PERCENTILES

• tell me what percentage of scores are less than your one.
• median = 50th percentile
• interquartile range = middle 50% of observations around the median
• to calculate percentile = (desired percentile/100) x (number of numbers + 1)

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:

1. 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
2. if the sample is randomly selected and large -> it can be assumed to be close to that of the population
3. the SD is used to calculate the standard error (see below)
4. 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

• small sample
• large variance within samples

CI vs P value

• p gives a probability of a specific hypothesis being right or wrong
• CI’s allow more scope for reader judgement on significance