Shapes and distribution PDF

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SP300: Shapes of Distributions Comparing measures of central tendency

Image: three examples of central tendency. The first is symmetrical, while the others show a clustering of scores et either end of the x axis. See slide 12 of last week’s lecture for more detail UK gross annual income • Median = £25948

Image: an example of a skewed distribution. The majority of incomes cluster around the lower end of the scale, meaning that the median is quite low relative to the full width of the scale (£0-250k+) Measures of dispersion: range and interquartile range • Range is very sensitive to outliers and skewed data • Interquartile range is less sensitive because we only look at the middle 50% of data

Image: a symmetrical (normal) distribution divided into quarters. The lowest and highest 25% of scores are excluded when considering the inter-quartile range. We can also call these percentiles, with the inter-quartile range being from the 25th to the 75th percentile Variance

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The average amount that each case in the sample differs from the mean Widely used in psychological statistics ´ = mean X N = number of cases ∑ = sum what follows

Computational formula for variance

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An alternative formula for working out variance by hand

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Gives the same result as the formula on the previous slide A useful shortcut but regular formula more instinctively expresses what variance is

Shapes of distributions • One thing you should always check is the way in which data on a particular variable is distributed • Many statistical tests assume a ‘normal distribution’ • From last week: shape of the distribution affects the mean/median and mode •

Major features of a distribution: • Central tendency • Variation • Skewness – whether the data is central, mostly to the left, or mostly to the right • Kurtosis – whether the distribution is especially flat or steep

Histograms • Histograms can vary in resolution

Image: The top image shows a steep curve, with large differences between column size. The lower image shows a smooth curve, with small differences between column size.

Frequency curve

Image: A histogram with a frequency curve. The frequency curve is a smoother illustration of the histogram which can make it easier to see the patterns in the data

Normal distribution

Image: A normal distribution curve. A normal distribution is symmetrical, so that data points cluster evenly about the mean, median, and mode, which will be the same number

Normal distribution (2) • It used to be thought that most characteristics in the natural world fit this pattern • Most statistical techniques have this as a basic assumption • But! You don’t need 100% normal distribution – an approximation is okay • Don’t forget: for normal distribution the mean, median and mode are equal Normal distribution: example

Image: overlapping frequency curves demonstrating height in men and women. Note that because they overlap, there are women with ‘male typical’ height, and men with ‘female typical’ height. Skewness •

How ‘lop-sided’ or asymmetrical the frequency curve is

Image: three bar charts showing negative skew (where scores cluster at the high end of the scale), a normal distribution, and a positive skew (where scores cluster at the low end of the scale)

Skewness (2)

• Negative skew: • More scores on the left of the mode than on the right • Mean and median are smaller than mode

Skewness (3) • Positive skew: • More scores on the right of the mode than on the left • Mean and median are larger than the mode

Kurtosis • Extent to which the frequency curve is excessively steep or flat

A flat (uniform) distribution

A steep distribution

Kurtosis (2) • Leptokurtic: A steep curve • Mesokurtic: A normal curve • Platykurtic: A flat curve

Bimodal distribution • Bimodal = 2 peaks • Multimodal = many peaks

• What variables might have a bimodal distribution?

Bimodal distribution (2) • Remember: the median and mean might not be at either peak

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Cumulative frequency distribution • Each point shows the number of cases that have that value or less • Should always be 100% at furthest right • Useful to look at, e.g. number of people falling below a target point

Percentiles • Percentile = % of cases that fall below a score in the distribution

• E.g. in a test you might rank people by percentile, i.e. how many did worse

Distribution shapes and data analysis...