Measures of Variability
31/12/10
FANZCA Part I Notes
- describes 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.
- to calculate = square the differences between each value and the sample mean, sum them
& divide this 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
Jeremy Fernando (2010)
sample mean represents a real trend in the sample.
(2) if the sample is randomly selected sample & large -> it can be assumed to be close to
that of the population.
(3) the SD is used to calculate the standard error (see later)
(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
= 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 & (b) small samples.
- SE used in parametric tests to quantify the difference between a sample mean & its
proposed population mean -> how far the two are apart in multiples of the SE (ztransformation)
- SE is used to calculate confidence intervals
CONFIDENCE INTERVALS
- 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
Jeremy Fernando (2010)
- 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.
Jeremy Fernando (2010)