13.8.4 Summary measures

Types of summary measure

• Frequency is the count of how often a value occurs.

• Relative frequency is that count as a proportion of the total, and cumulative frequency is a running total of the counts.

• The type of data limits the choice: a single Likert item gives ordinal data, so it is summarised by the median and mode, not the mean and standard deviation.

Worked dataset

Lengths of stay for 15 patients, in days: 3, 5, 7, 8, 10, 12, 14, 16, 19, 23, 28, 34, 47, 96, 188.

Mean

• The arithmetic mean is the sum of all values divided by the number of values, the full count n and not n minus 1: here 510 / 15 = 34 days.

• The mean is the balance point of the data: the deviations from it sum to zero, so the total distance below equals the total distance above.

• The mean uses every observation, so changing any one value changes it, whereas the median, mode, range and interquartile range can stay the same.

• One extreme value shifts the mean by an amount that grows with the size of the outlier: removing the 188-day stay drops the mean from 34 to 23 days (322 / 14).

• Adding a constant to every value shifts the mean and other measures of location by that constant and leaves measures of spread unchanged.

• The mean multiplied by the number of discharges gives total bed use.

Median

• The median is the positional centre of ranked data, with equal numbers of observations above and below it.

• With an odd number of values it is the single middle value, at position (n + 1) / 2: the 8th of 15 stays, 16 days.

• With an even number it is the mean of the two central values, so it may be a value nobody had: without the 188-day stay, the median of 14 stays is (14 + 16) / 2 = 15 days.

• The median depends only on rank order, so adding an outlier moves it by one rank position at most.

• This makes the median the preferred measure of central tendency for skewed data, because the mean is pulled towards the tail.

• The line inside the box of a box-and-whisker plot marks the median.

The median has equal numbers of observations on each side; the mean has equal total distance on each side.

Mode

• The mode is the most frequently occurring value, found by counting repeats.

• The mode is the measure of central tendency for nominal or categorical data, because it needs only the count in each category.

• When no value repeats, as in the length-of-stay data, there is no unique mode.

• An outlier leaves the mode unchanged unless it repeats an existing value.

• A distribution with two separate peaks is bimodal, and tied peaks are reported as more than one mode, never averaged into one value.

• Multiple peaks can suggest a mixture of populations but do not prove it.

Range, percentiles and quartiles

• The range is the largest value minus the smallest: 188 minus 3 is 185 days.

• The range rests on the two extreme values alone, so one extreme value inflates it, and it tends to widen as the sample grows.

• A percentile rank describes a value's relative standing within the data, not a raw percentage of a score.

• A quartile is found as the median of the lower or upper half of the ordered data.

• The interquartile range is Q3 minus Q1, the 75th minus the 25th percentile: 34 minus 8 is 26 days.

• The interquartile range spans the middle half of the data, so 50% of values lie between Q1 and Q3.

• The interquartile range is rank-based and distribution-free, and extreme values do not affect it, so it is the measure of spread that goes with the median for skewed data.

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Variance and standard deviation

• Sample variance is built from the squared deviation of every observation from the sample mean, summed and averaged, with no square root taken.

• For a sample, the sum of squared deviations is divided by n minus 1 (Bessel's correction) to give an unbiased estimate of the population variance; a complete population is divided by n.

• The standard deviation (SD) is the positive square root of the variance, so the variance is the SD squared.

• The SD keeps the units of the raw data, while the variance is in squared units. Neither can be negative.

• When all values are identical there is no dispersion, and the variance and SD are both zero.

• The SD is measured around the mean, so the mean is a necessary input to it.

• Every value's squared distance from the mean enters the SD, so it is sensitive to outliers: the 188-day stay supplies about 70% of the squared deviations behind an SD of about 49 days.

• The coefficient of variation is the SD divided by the mean, usually as a percentage. It has no units, so it compares variability across variables on different scales or with different means.

Standard deviation and standard error

• The SD describes how much individual patients vary; the standard error (SE) describes how precisely a sample estimate is known.

• For the mean of an independent sample, SE = SD / square root of n: with an SD of 49, n = 15 gives 12.6 days and n = 150 gives 4.0 days.

• More observations make the estimate more precise without making individuals less variable, so error bars must say which measure they show.

Normal distribution

A normal distribution is symmetrical about a single peak, so its mean, median and mode coincide.

• This is the empirical rule. A stated range must first be converted into a number of SDs from the mean.

• The slice between the mean and 1 SD on one side holds about 34%.

• The curve is symmetrical, so the share outside a band splits equally between the two tails: about 16% lies beyond 1 SD on one side and about 2.5% beyond 2 SD.

• The share below a cut-off is the central band plus the opposite tail: 68% plus 16% puts 84% below +1 SD, and 95% plus 2.5% puts 97.5% below +2 SD.

• The central limit theorem states that the sampling distribution of the mean becomes approximately normal as sample size grows, whatever the shape of the parent population.

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Skewed distributions

Skew is named by the side of the long tail. The mean is pulled furthest into the tail, the mode stays at the peak and the median lies between them.

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• The order reads off the direction: mean above median above mode is positive skew, and the reverse is negative skew.

• The length-of-stay data are positively skewed: the mean of 34 sits above the median of 16, and 12 of 15 stays (80%) end by day 34.

• A similar mean and median suggest symmetry but do not establish it.

Choosing summaries for the data

• Normal continuous data are summarised by the mean and SD, and skewed data by the median and interquartile range.

• A mean smaller than the SD, for a measure that cannot be negative, shows the data are positively skewed rather than normal.

• Applying the empirical rule to the length-of-stay data shows the error: the mean minus two SDs is 34 minus 98, an impossible stay of -64 days.

Proportions, rates and odds

• A proportion divides a subset by its total and lies between 0 and 1: 40 detained admissions among 200 is 0.20, or 20%.

• A rate includes person-time: 18 readmissions over 100 patient-years is 0.18 per patient-year.

• A percentage labelled "readmission rate" is usually a risk over a fixed period, not a person-time rate.

• Odds are events relative to non-events, odds = p / (1 - p): a risk of 0.5 gives odds of 1, and a risk of 0.9 gives odds of 9.

• Response risks of 70% and 50% give a risk ratio of 1.4 but an odds ratio of 2.33, so an odds ratio must not be described as a risk ratio.