*Reviewed and revised 26 August 2015*

**OVERVIEW**

- Measures if 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

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