Learn — Descriptive statistics

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Central tendency — mean, median and mode

A measure of central tendency is a single number that stands for the typical or central value of a set of observations, and the three in use — the mean, the median and the mode — answer that question in three different ways: the arithmetic average, the middle value and the most frequent value. Which one to report depends on the level of measurement and on the shape of the distribution, and examinations test both the calculation and the choice.

The mean is the sum of the values divided by their number — 10, 20, 30, 40 and 50 give 150 ÷ 5 = 30, and five haemoglobins of 9, 11, 12, 10 and 13 g/dL give 55 ÷ 5 = 11 g/dL; it uses every value, is the most stable from sample to sample and is the basis of the parametric tests, but it is pulled towards extreme values and needs interval or ratio data; the sum of the deviations from the mean is always zero.
The median is the middle value when the observations are arranged in order — of 3, 5, 7, 9 and 11 it is 7 — and with an even number it is the average of the two middle values, so pulse rates of 68, 72, 76, 80, 84 and 90 give (76 + 80) ÷ 2 = 78; it is the 50th percentile, it is unaffected by outliers, and it is the measure for skewed data such as length of stay or income and for ordinal data.
The mode is the value that occurs most often — of the blood groups A, O, B, O, AB, O, A and B the modal group is O — and it is the only measure of central tendency for nominal data; a distribution may be bimodal or have no mode, and the mode is the least stable of the three.
The choice follows the level and the shape: nominal data take the mode, ordinal data the median, symmetrical interval or ratio data the mean, and skewed interval or ratio data the median; the mean is reported with the standard deviation and the median with the range or the interquartile range.
In a symmetrical, normal distribution the mean, the median and the mode are equal; in a positively (right) skewed distribution with a few very large values — hospital length of stay — the mean is greater than the median, which is greater than the mode; in a negatively (left) skewed one the order reverses, and the mean is always the one dragged towards the tail.
For grouped data in a frequency table the mean is the sum of each class frequency times its midpoint divided by the total, the median lies in the class where the cumulative frequency passes half, and the mode in the class with the highest frequency; the geometric and harmonic means serve rates and ratios.

Dispersion — range, variance, standard deviation and quartiles

A measure of dispersion (variability or spread) says how far the observations are scattered around their centre, and it is reported beside the measure of central tendency because two wards can share a mean length of stay and differ entirely in how much their patients vary. The measures are the range, the interquartile range, the variance and the standard deviation, and the standard deviation is the one that goes with the mean and with the normal curve.

The range is the highest value minus the lowest — 12, 18, 25, 30 and 45 give 45 − 12 = 33; it is the simplest and the crudest measure, uses only two values, and is inflated by a single outlier.
A percentile is the value below which a given percentage of the observations fall, and the quartiles divide the ordered data into four — the first quartile is the 25th percentile, the second the median, the third the 75th; the interquartile range, the third quartile minus the first, holds the middle half of the values and is the spread reported with the median for skewed or ordinal data, which the box-and-whisker plot draws.
The variance is the mean of the squared deviations from the mean — the sum of the squared deviations divided by n for a population and by n − 1 for a sample — and because it is in squared units it is not read directly but taken as the step to the standard deviation.
The standard deviation is the square root of the variance, in the same units as the data: for the population 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5, the squared deviations add to 32, 32 ÷ 8 = 4 and its square root is 2; a small standard deviation means the values cluster near the mean, a laboratory value is called normal when it lies within about two standard deviations of the mean, and the sample formula divides by n − 1.
The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage — a unit-free measure for comparing the spread of variables measured in different units or with very different means, such as weight against height.
The standard error of the mean, the standard deviation divided by the square root of n, is not a measure of the spread of individuals but of how much sample means would vary around the population mean; it shrinks as the sample grows while the standard deviation does not, and it belongs to inference and confidence intervals rather than to description.

Normal distribution — the empirical rule, skewness and the mean–median–mode relation

The normal (Gaussian) distribution is the symmetrical, bell-shaped curve that many continuous biological measurements — height, haemoglobin, blood pressure in a healthy population — approximate, described completely by two numbers, its mean and its standard deviation. It underlies the definition of a normal laboratory range, the z-score and every parametric test, and the examinable facts are its shape, the proportion of values within one, two and three standard deviations, and what happens to the mean, median and mode when the curve is skewed.

The curve is bell-shaped, symmetrical about the mean and unimodal, its mean, median and mode coincide at the centre, its tails approach the baseline without ever touching it, the total area under it is 1, and it is fixed entirely by its mean and its standard deviation.
The empirical rule: about 68 per cent of values lie within one standard deviation on either side of the mean, about 95 per cent within two (more exactly 1.96) and about 99.7 per cent within three — so 2.5 per cent lie beyond two standard deviations at each end and 16 per cent above one; a laboratory reference range is the mean plus or minus two standard deviations of a healthy population, which is why one healthy person in twenty falls outside it.
The standard normal curve has a mean of 0 and a standard deviation of 1, and a z-score, the value minus the mean divided by the standard deviation, says how many standard deviations a value lies from the mean — a haemoglobin of 9 in a population with mean 13 and standard deviation 2 has z = −2; a z beyond ±1.96 lies outside the central 95 per cent, which is the two-tailed 5 per cent level.
A positively (right) skewed curve has its long tail to the right — a few very large values, such as hospital length of stay, income or parity — and its mean is greater than its median, which is greater than its mode; a negatively (left) skewed curve has the tail to the left and the order reversed; the mean is always pulled towards the tail, so the median is the average to report, and kurtosis describes how peaked the curve is.
The parametric tests — t, analysis of variance, Pearson's r — assume normally distributed interval or ratio data, and the central limit theorem, which says that the means of repeated samples are normally distributed whatever the population's shape once the sample is about thirty or more, is why the mean can be tested at all; badly non-normal data take a transformation or a non-parametric test.
The other distributions have their uses — the binomial for dichotomous outcomes, the Poisson for rare events, the t-distribution for small samples with its wider tails — and a histogram or a normality test such as Shapiro–Wilk checks the shape of the data before the test is chosen.

Data presentation — frequency tables, histograms, bar and pie charts, percentiles

Data presentation is the arrangement of raw observations into tables and graphs so that their pattern can be seen at once — the frequency of each value, the shape of the distribution, the comparison between groups and the change over time. The rules are few: the right form for the level of the data, a title and labelled axes, and no chart that shows a difference the numbers do not.

A frequency distribution table lists the values or class intervals in one column and the number of observations in the next, with the relative frequency as a percentage and the cumulative frequency beside them; class intervals are of equal width, mutually exclusive and exhaustive, usually five to ten of them, and the table is the first step of every analysis and the basis of the median class and the histogram.
Quantitative (interval or ratio) data take the histogram — adjacent bars whose area is proportional to frequency, for continuous data grouped in class intervals, which shows whether the shape is normal or skewed — the frequency polygon that joins the midpoints and lets two distributions share one chart, the ogive or cumulative frequency curve from which the median and the percentiles are read, the line graph for a variable over time such as a temperature chart, and the scatter diagram for two continuous variables.
Qualitative (nominal or ordinal) data take the bar chart — separate bars of equal width with gaps between them, height equal to frequency, for categories such as blood groups or wards — and the pie chart, whose sectors are proportional to each category's share of the whole, best for a few categories such as causes of death; the gap between the bars is the tell that separates a bar chart from a histogram.
A percentile is the value below which a given percentage of the observations lie — a child at the 10th percentile for weight is heavier than 10 per cent of children of that age — the growth chart's 3rd, 50th and 97th centiles are the ones nurses read, the 50th percentile is the median, and deciles and quartiles cut the data into tenths and quarters.
The box-and-whisker plot shows the median, the quartiles and the extremes in one figure and exposes skewness and outliers, and the stem-and-leaf display keeps the raw values while showing the shape.
Every table and figure is numbered and titled, the axes are labelled with their units, a bar chart starts at a zero baseline because a truncated axis exaggerates a difference, the sample size and the source are stated, a percentage is given with its denominator, and a figure never merely repeats a table.

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