How do you summarise a set of numbers?

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What does a column of numbers tell you?

A list of 40 hospital stays is hard to read. Descriptive statistics reduce it to a few numbers: a typical value, how much the values vary, and the shape of the distribution.

The typical value is a measure of central tendency (mean, median, mode). The variation is a measure of dispersion (range, interquartile range, variance, standard deviation). The shape is described by skewness and kurtosis. Draw the histogram before you choose any of them. How to draw it is in the data presentation chapter.

Central tendency

What is the typical value: mean, median or mode?

The mean is the sum of the values divided by their number. The median is the middle value when the values are put in order. The mode is the most frequent value.

A distribution with two peaks is bimodal. It often means two different groups have been mixed, such as the heights of men and women together. Describe the groups separately, because a single mean or median falls between the two peaks and describes neither group. The mode is also unstable in small samples, where one or two extra values can move it.

Interactive

Move one patient's stay and watch the mean, the median and the SD

Teaching example: six patients stay 2, 3, 3, 3, 4 and 5 days. Move the seventh patient's stay and watch both averages.

With a seventh stay of 6 days: mean 3.7, median 3, SD 1.4 days. With 40 days: mean 8.6, median still 3, SD 13.9 days. One extreme value changes the mean and the SD. The median does not change, because it depends only on which value is in the middle.

Central tendency

When is the ordinary mean the wrong average?

When values rise in multiples, when you average rates, and when you combine groups of different sizes.

Geometric mean: values that rise in multiples

GM = antilog of the mean of the logs = (x₁ × x₂ × … × xₙ)^1/n

Teaching example: antibody titres in seven people after vaccination, 1:20, 1:40, 1:40, 1:80, 1:160, 1:320 and 1:640. Each dilution step doubles the titre, so the values are analysed on a log scale. The mean of the log₂ values is 6.608, and 2^6.608 = 97.5, a geometric mean titre (GMT) of about 1:98. The arithmetic mean, 185.7, is pulled up by the two highest titres. The median is 80. The geometric mean cannot be calculated if any value is zero or negative.

Harmonic mean: averaging rates

HM = n ÷ Σ(1 ÷ x)

Teaching example: two pharmacists each dispense 60 prescriptions, one at 20 per hour and one at 30 per hour. Together they take 3 + 2 = 5 hours for 120 prescriptions, which is 24 per hour. The harmonic mean gives the same answer, 2 ÷ (1/20 + 1/30) = 24, while the ordinary mean of the two rates, 25, is wrong. The harmonic mean suits rates when the same amount of work is done at each rate.

Combined mean of groups of different sizes

Combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂)

Teaching example: ward A, 40 patients, mean stay 6.0 days. Ward B, 60 patients, mean stay 4.5 days. Combined mean = (240 + 270) ÷ 100 = 5.1 days. The average of the two means, 5.25, gives the smaller ward too much weight. Medians cannot be combined in this way.

Dispersion

How much do the values vary?

Two groups can have the same mean and be very different. Measures of dispersion describe how widely the values are spread.

Why square the deviations? Deviations from the mean always add up to zero, so their signs must be removed. Absolute values do this, and give the mean deviation. Squares are used for the variance because variances add: for two independent measurements, the variance of their sum or difference is the sum of their variances. If two independent readings each have an SD of 3 mmHg, the difference between them has an SD of √(3² + 3²) = 4.24 mmHg, not 6 and not 0.

Notation: μ and σ for the population mean and SD, x̄ and s for the sample mean and SD. σ² and s² are the variances.

The formula

How do you calculate the mean, SD and CV?

Teaching example: haemoglobin in six women, 10, 11, 12, 12, 13 and 14 g/dL.

mean (x̄) = Σx ÷ n

SD = √[ Σ(x − x̄)² ÷ (n − 1) ]

CV = SD ÷ x̄ × 100  ·  SE = SD ÷ √n

x

each value

n

number of values

Σ

add up

n − 1

used in a sample, because deviations from the sample mean are slightly smaller than deviations from the population mean

x̄ = 72 ÷ 6 = 12.0 g/dL

variance = 10 ÷ 5 = 2.0 → SD = √2.0 = 1.41 g/dL

CV = 1.41 ÷ 12.0 × 100 = 11.8%  ·  SE = 1.41 ÷ √6 = 0.58 g/dL

The same calculation in Python:

import statistics as st

hb = [10, 11, 12, 12, 13, 14]    # g/dL, six women (teaching example)
mean = st.mean(hb)
sd = st.stdev(hb)                # divides by n - 1
cv = sd / mean * 100
se = sd / len(hb) ** 0.5
print(f"mean {mean:.1f}, SD {sd:.2f}, CV {cv:.1f}%, SE {se:.2f}")

# mean 12.0, SD 1.41, CV 11.8%, SE 0.58

Median 12, mode 12, range 10–14, IQR 11.25–12.75 (linear interpolation), mean deviation 1.0 g/dL. Dividing by n instead of n − 1 gives 1.29, which underestimates the population SD.

What happens when every value is changed in the same way?

Adding a constant moves the centre but not the spread. Multiplying changes both by the same factor. The CV changes with the origin, which is why it is used only for measurements with a true zero.

The formula

How do you get the mean, median and mode from a frequency table?

When only grouped data are available, each value is assumed to lie spread evenly within its class interval. The results are approximations.

Teaching example (exam-style, made-up): systolic blood pressure of 150 adults. 100–110 means 100 to under 110 mmHg.

Mean = Σfx ÷ n = 19,790 ÷ 150 = 131.9 mmHg

Median = L + [(n/2 − cf) ÷ f] × h = 130 + [(75 − 71) ÷ 38] × 10 = 131.1 mmHg

Mode = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h

Mode = 120 + [(41 − 22) ÷ (82 − 22 − 38)] × 10 = 128.6 mmHg

L

lower limit of the class that contains the median (or the mode)

cf

cumulative frequency of the classes before it

f

frequency of that class. For the mode, f₁ is the largest frequency, f₀ and f₂ the frequencies of the classes before and after

h

width of the class, here 10 mmHg

Quartiles use the same formula as the median, with n/4 and 3n/4 in place of n/2. Q1 = 120 + [(37.5 − 30) ÷ 41] × 10 = 121.8, Q3 = 140 + [(112.5 − 109) ÷ 24] × 10 = 141.5 mmHg. The SD from the midpoints is 14.3 mmHg. With the individual values, always calculate from them instead.

Comparison

SD or SE: which one goes after the ± sign?

The SD describes how much patients differ from each other. The standard error (SE) describes how much the mean would differ from study to study if you repeated the study many times.

Back to the thesis table: an SE of 1.1 with 40 patients means SD = 1.1 × √40 = 7.0 days. Mean − 2 SD is 6.2 − 13.9 = −7.7 days, which is impossible for a hospital stay. The data are right-skewed, so the table should give the median and IQR.

Interactive

Change the number of patients and watch the SD and the SE

The SD describes how much patients differ from each other. The SE describes how much the mean differs between samples of the same size, and it becomes smaller as the sample size increases.

Laboratory imprecision

How do you compare spread when the means differ?

An SD is in the units of the measurement and depends on the size of the values. Divide it by the mean and it has no units.

CV (%) = SD ÷ mean × 100

In a laboratory, imprecision is measured by analysing the same control material repeatedly and calculating the SD and CV. The CV lets you compare analytes with different units, one analyte at different concentrations, and one method between laboratories. Laboratories then compare the CV with an analytical performance specification. A 2015 European Federation of Clinical Chemistry and Laboratory Medicine (EFLM) consensus described three models for setting these specifications.

Use the CV only for measurements with a true zero (concentrations, weights). It has no meaning for temperature in °C, and it is unstable when the mean is close to zero.

The normal curve

What is a normal distribution?

A symmetrical, bell-shaped distribution fixed completely by its mean and its SD. Mean, median and mode are equal, and the tails extend without ever reaching zero.

z = (x − mean) ÷ SD

The standard normal distribution has mean 0 and SD 1. The z-score tells you how many SDs a value lies from the mean, so values measured in different units can be placed on one scale. It is called the Gaussian distribution after Carl Friedrich Gauss. The 95% rule is used for reference ranges (mean ± 1.96 SD) only when the data are close to normal in shape.

For a distribution of any shape, a weaker statement holds: at least 1 − 1/k² of the values lie within k SDs of the mean. That is at least 75% within 2 SD and 88.9% within 3 SD. This is Chebyshev's inequality, also called the Bienaymé–Chebyshev rule. For a normal distribution the figures are 95.4% and 99.7%.

Percentiles

What does '3rd percentile' mean on a growth chart?

A percentile is the value below which a given percentage of values lie. The 3rd percentile has 3% of values below it.

Percentiles do not need a normal distribution. They are read from the ordered data, which is why growth charts, reference ranges for skewed tests and the IQR use them.

Percentiles belong to a family of quantiles, values that cut ordered data into groups of equal size: 2 tertiles make 3 groups, 3 quartiles make 4, 4 quintiles make 5, 9 deciles make 10 and 99 percentiles make 100. A percentile is not a percentage. A child on the 90th percentile of weight is heavier than 90% of children of the same age and sex on that chart, whatever the weight.

A median is a statement about a group. A median survival of 3 years means that half the patients lived 3 years or less and half lived longer. It does not predict 3 years for any one patient.

Interactive

Move a value along the normal curve and watch its z-score and percentile

On a normal curve each z-score corresponds to one percentile. The shaded area is the share of values below the value, above it, or no further from the mean than it.

Shape

What are skewness and kurtosis?

Skewness describes whether one tail of the distribution is longer than the other. Kurtosis describes the peak and tails compared with a normal curve of the same SD.

Pearson's skewness = (mean − mode) ÷ SD  or  3 × (mean − median) ÷ SD

For the hospital-stay curve at the top of this chapter (SD 3.02 days): (5.67 − 3.89) ÷ 3.02 = 0.59, and 3 × (5.67 − 5.00) ÷ 3.02 = 0.66. Positive values mean right skew. Zero means symmetry.

Quartile coefficient of skewness = (Q3 + Q1 − 2 × median) ÷ (Q3 − Q1)

It uses only the quartiles, so extreme values do not affect it. For the same curve, Q1 = 3.57, median 5.00 and Q3 = 7.01 days: (7.01 + 3.57 − 10.00) ÷ (7.01 − 3.57) = 0.17. The value lies between −1 and +1. Software usually gives a moment coefficient instead, based on the average cube of the deviations (m₃ ÷ m₂^1.5). The different coefficients are not interchangeable, so report which one you used.

The order mode < median < mean in a right-skewed distribution is a rule of thumb, not a law. It fails most often in discrete distributions, and can fail in distributions with more than one peak or with one long tail and one heavy tail (von Hippel, 2005).

β₂ = m₄ ÷ m₂²  ·  excess kurtosis = β₂ − 3

m₂ and m₄ are the average second and fourth powers of the deviations from the mean. m₂ is the variance. Some programs report β₂ and others report excess kurtosis, β₂ − 3, so check which one your software gives before you interpret it.

Interactive

Lengthen the right tail and watch the mean, median and mode separate

In a right-skewed distribution the mode is smaller than the median and the median is smaller than the mean. The longer the tail, the larger the gaps between them.

Shape

How do you check whether your data are close to normal?

Look at the data first. Use a formal test as a supplement to the plots, not a replacement for them.

1. Draw a histogram or a dot plot. Look for one peak and rough symmetry.
2. Compare the mean and the median. For a value that cannot be negative, check whether mean − 2 SD falls below zero.
3. Draw a box plot. The median should lie near the middle of the box, with whiskers of similar length.
4. Draw a normal Q–Q plot. The points should lie close to the straight line.
5. If needed, add a formal test such as the Shapiro–Wilk test. With small samples such tests often fail to detect a real departure, and with large samples they flag departures too small to matter.

Right-skewed data can often be analysed after a log transformation, with results summarised as geometric means, or with non-parametric methods (see choosing a test). Q–Q plots, residual checks and transformations are covered in detail in Checking assumptions.

Reference intervals

What is a normal range, and how is it set?

A reference interval is the range that contains the central 95% of values in healthy reference people. By definition, 5% of healthy people have a value outside it.

Two ways to calculate it. If the values, or their logarithms, are close to normal, use mean ± 1.96 SD. Otherwise use the 2.5th and 97.5th percentiles of the ordered values. The Clinical and Laboratory Standards Institute (CLSI) C28-A3 guideline recommends the non-parametric (percentile) method, and at least 120 reference individuals so that confidence intervals for the two limits can be calculated (Ozarda, 2016).

A result outside the reference interval is not proof of disease, and a result inside it is not proof of health. The more tests are done, the more healthy people have at least one result outside:

P(at least one outside) = 1 − 0.95^k

With k independent tests: 5 tests 22.6%, 10 tests 40.1%, 20 tests 64.2%.

This assumes the tests are independent. Related tests, such as liver enzymes, are not, so the true figure differs. Establishing reference intervals in a laboratory, and comparing a result with the same patient's earlier results, are covered in Laboratory quality control and reference intervals. Choosing a diagnostic cut-off is covered in the diagnostic tests chapter.

Interactive

Lengthen the tail and add tests, and watch how many healthy people fall outside the normal range

Distributions

Which named distributions will you meet?

A distribution describes how often each value occurs. A few named distributions appear again and again in medical statistics.

Binomial and Poisson distributions are discrete: only whole numbers are possible. The others are continuous. The t, chi-square and F distributions are used mainly to calculate p-values, in the hypothesis testing chapter. Counts and rates with a Poisson distribution are modelled in Poisson and log-linear models.

Categorical data

How do you summarise categories?

Give the number and the percentage in each category, and say what the percentage is out of.

Percentage points or per cent? A rise from 20% to 25% is an increase of 5 percentage points, and a relative increase of 25%. Write which one you mean. Averages for categories: the mode is the only average for nominal data (blood group). For ordered categories (cancer stage) the median category can also be given.

Incidence, prevalence and the rates used in epidemiology are covered in the risk and survival chapter. Tables of questions that allow more than one answer, where the percentages add up to more than 100, are covered in the data presentation chapter.

Comparison

Are you describing your patients, or drawing conclusions about all patients?

Descriptive statistics summarise the data you collected. Inferential statistics use that sample to draw conclusions about the population, with a stated uncertainty.

95% CI = 12.0 ± 2.571 × 0.58 (t with 5 degrees of freedom). Tests and confidence intervals are covered in the hypothesis testing chapter.

Decide

Which summary should you report?

Answer two or three questions about each variable.

By specialty

Descriptive statistics problems by specialty

Common myths

Common mistaken beliefs

• ✗Myth: The SEM is fine for error bars and makes the data look tidy.

  ✓The SEM describes the precision of the mean, not the spread of patients, and it is always smaller than the SD. A review of four anaesthesia journals from 2001 found this misuse in 198 of 860 articles (23%) (Nagele, 2003).

• ✗Myth: The SD can be used only for normally distributed data.

  ✓The SD is a valid measure of variability whatever the distribution (Altman and Bland, 2005). The statement "about 95% of values lie within 2 SD" is true only for data close to normal in shape.

• ✗Myth: The mean is always the best average.

  ✓With skewed data or extreme values, use the median. One stay of 40 days raises the mean of seven stays from 3.7 to 8.6 days. The median stays at 3.

• ✗Myth: A "normal distribution" means the values of healthy people.

  ✓It is the name of one bell-shaped distribution. Values from healthy people can be skewed, and values from patients can be normal in shape.

• ✗Myth: Enrolling more patients makes the SD smaller.

  ✓The SD estimates the spread in the population, so it stays about the same. The SE becomes smaller (SE = SD ÷ √n).

• ✗Myth: A smaller SD always means a more precise test.

  ✓Only when the means are similar. Glucose SD 12 at mean 101 (CV 11.9%) is more precise than creatinine SD 0.5 at mean 1.2 (CV 41.7%).

Why this matters to you

Five consequences of getting descriptive statistics wrong

1. A protocol that does not match the data

   The analysis plan says "all data will be expressed as mean ± SD", and then hospital stay, drain output and costs turn out to be skewed.

2. Thesis figures that hide the variation

   SEM error bars look narrow and suggest that patients responded alike when they did not.

3. A common viva question

   Examiners ask the difference between SD and SE, and why you chose the mean or the median for each variable.

4. Misreading papers

   A stay reported as 4 (5) days with mean − 2 SD below zero is skewed. The mean overstates the typical stay.

5. Laboratory results that change with repetition

   With creatinine SD 0.5 mg/dL at a mean of 1.2, about half of repeat pairs on the same sample differ by more than 0.5 mg/dL, enough to move a patient across a clinical cut-off.

History

History of descriptive statistics

1. 1733

   Abraham de Moivre

   In a Latin pamphlet, gave an approximation to the binomial distribution for a large number of trials. The expression is the one now recognised as the normal curve. Historians disagree about whether this counts as discovering the normal distribution.

2. 1809

   Carl Friedrich Gauss

   In Theoria motus corporum coelestium, a book on calculating planetary orbits, presented the method of least squares with the normal curve as the distribution of errors of observation. The normal distribution is also called the Gaussian distribution.

3. 1893–94

   Karl Pearson

   Introduced the term "standard deviation" in 1893 and used it in print in 1894, where he equated it with Gauss's "mean error". In the same 1894 paper he called the error curve the "normal curve".

4. 1895

   Karl Pearson

   In a paper on skew variation, used the term "mode" for the value of maximum frequency and measured skewness from the distance between the mean and the mode.

5. 1896

   Karl Pearson

   Defined the coefficient of variation: the standard deviation divided by the mean, multiplied by 100.

6. 1905

   Karl Pearson

   In Biometrika, called β₂ − 3 "the degree of kurtosis" and used the terms platykurtic, leptokurtic and mesokurtic.

7. 1918

   Ronald A. Fisher

   Named the square of the standard deviation the "variance", in a paper on inheritance.

8. 2003

   Peter Nagele

   Found the SEM used to describe sample variability in 198 of 860 articles (23%) in four anaesthesia journals published in 2001.

9. 2005

   Douglas Altman and Martin Bland

   BMJ statistics note "Standard deviations and standard errors": the two are often confused, and the SD is a valid measure of variability regardless of the distribution.

Guidelines

Descriptive statistics in reporting guidelines

Reading a paper

Five checks on any paper's descriptive statistics

• Does the paper say whether each number after ± or in brackets is an SD, an SE or a confidence interval?
• Are skewed variables (hospital stay, costs, many laboratory values) given as median and IQR?
• For a value that cannot be negative, is mean − 2 SD below zero? If so, the data are skewed.
• Are error bars in the figures labelled, and do they show SD when they describe patients?
• Are the summaries given for each group, with the number of patients in each?

Why is it called that?

Word surgery

Exam answers

Model answer outlines

Past papers

Every TNMGRMU question on descriptive statistics

Each one is answered in this chapter. Tap a sitting to open the university's own question paper.

1. ×4Standard DeviationDM Medical Oncology · Feb 2023MD Biochemistry · Oct 2015MS OBSTETRICS AND GYNAECOLOGY · Oct 2015MD Community Medicine · Oct 2013
2. ×3Measures of DispersionMD Biochemistry · May 2023Ph.D. RM · Aug 2021MD Community Medicine · May 2018
3. ×2Standard error and standard deviationMD Community Medicine · May 2025Ph.D. RM · Jul 2025
4. ×2Standard Error & Standard DeviationPh.D. RM · Jan 2025Ph.D. RM · Jul 2024
5. ×2Laboratory results for glucose shows a standard deviation of 12, mean of 101 mg/dL, whereas for creatinine the observed standard deviation is 0.5 and mean is 1.2 mg/dL. The quality manager judges analytical performance for glucose to be better than that of creatinine. JustifyMD Biochemistry · May 2023MD Biochemistry · May 2011
6. ×1Principle and types of ANOVA (analysis of variance)MD Biochemistry · Oct 2025
7. ×1Write a note on the measures of central tendency and dispersion in biostatisticsDM Pulmonary Medicine · Aug 2025
8. ×1Mode and medianPh.D. RM · Jan 2023
9. ×1Median and meanPh.D. RM · Jul 2022
10. ×1Explain ‘Skewed and normal distribution of data’ with relevant examplesMD Community Medicine · May 2019
11. ×1Define the common measures of central tendency for data setPh.D. RM · Jan 2019
12. ×1Mean, mode, median and standard deviationPh.D. RM · Jul 2019
13. ×1Explain mean, standard deviation and coefficient of variation in medical practice with appropriate examplesPh.D. RM · Jul 2018
14. ×1Measures of central tendency and their usesMD Community Medicine · May 2017
15. ×1Concept of standard deviation and percentilesDM Neonatology · Feb 2017
16. ×1Normal distribution (Gaussian curve)DM Neonatology · Aug 2017
17. ×1Write in brief about (a) Mean (b) median and (c) modePh.D. RM · Jul 2017
18. ×1Explain in brief about skewness and kurtosisPh.D. RM · Jul 2017
19. ×1Explain standard deviation and standard error meanPh.D. RM · Jul 2017
20. ×1It is better to express imprecision of an analytical method in terms of coefficient of variation rather than standard deviation. Justify with an exampleMD Biochemistry · Oct 2014
21. ×1Statistical significance of Standard deviationMD Biochemistry · Apr 2013
22. ×1Define normal distributionMD Pharmacology · May 2011
23. ×1Define standard deviationMD Pharmacology · May 2011
24. ×1Correlation coefficient and coefficient of variationMD Biochemistry · Sep 2009
25. ×1Standard deviationsDM Medical Oncology · Oct 2000
26. ×1Calculation of standard deviation and t-value of small experimental groups. ]MD Biochemistry · Apr 1994
27. ×1Write brlefly an : (a) Analytical epidemiology. (b) Standard Deviation. (c) Vitamln A Deficiency. (d) Dimehslons of growth and developmentMD Community Medicine · Mar 1990

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