← Common Core coursesBMS 1110 · Applied Mathematics and Medical Biostatistics
BMS 1110 · Common Core (Tronc commun)Section 4.5 · Organising and describing data
Section 21 of 30

Mean, median and mode

Compute the mean, median and mode of a small set of clinical observations, and choose the measure that describes the data honestly when the values are uneven.

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  1. Learn gives you one short idea at a time. Each idea starts with a single sentence, then explains it, gives an example, and asks you a small question you can answer straight away.
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  4. “Go deeper” is optional. It adds mechanism and extra detail for when you are ready — nothing you need for the checks is hidden there.
Why this matters

Almost every clinical summary a nurse meets is one of these three numbers. A ward’s average length of stay, a clinic’s typical waiting time, the most common blood group on a unit — each is a mean, a median or a mode, and each answers a different question. Knowing which one is being used, and which one ought to be used, is what stops a single unusual patient from being presented as if they were typical.

What you need to know first

You need only two habits you already have: adding a short column of numbers, and dividing one whole number by another. You should also be able to put a list in order from smallest to largest. Nothing here needs a calculator, and every number below is small enough to check by hand.

Words to know (5)
Mean
The arithmetic mean, or average, is the total of all the observations divided by the number of observations. It keeps the unit of the data. Pulse rates 72, 80, 76, 84 and 78 total 390, and 390 ÷ 5 = 78, so the mean is 78 beats per minute.
Median
The median is the middle observation once the data are ordered from smallest to largest. With an even number of observations it is the mean of the two middle values. Ordered values 5, 7, 8, 12 and 15 have median 8, because 8 is the third of five values.
Mode
The mode is the value that occurs most frequently. A dataset may have no mode, one mode, two modes (bimodal) or several. In 2, 3, 3, 4, 5, 3 and 6 the value 3 appears three times, so the mode is 3.
Outlier
An outlier is one observation that lies far away from the rest of the data. It changes a mean much more than it changes a median. In the waiting times 12, 14, 15, 15, 17, 18 and 90 minutes, the 90 is an outlier.
Central tendency
Central tendency is the general name for the mean, the median and the mode together, because each one describes where the centre of the data lies. A ward summary that gives “mean 78, median 78, mode 72 beats per minute” reports three measures of central tendency.
One idea at a timeIdea 1 / 5

In one sentence The arithmetic mean, or average, is the total of every observation divided by the number of observations, and it keeps the unit of the data.

The mean adds every observation and divides by how many there are

The arithmetic mean — the number most people simply call the average — is the total of all the observations divided by the number of observations. In words, the rule is: mean = total of every value ÷ n, where n is how many values there are. Statisticians write the sample mean as x̄, read “x bar”, which is just a short label for this calculation. The notes’ five pulse rates are 72, 80, 76, 84 and 78 beats per minute. Adding them gives 72 + 80 + 76 + 84 + 78 = 390. There are n = 5 observations, so the mean is 390 ÷ 5 = 78 beats per minute. Two things are worth noticing. First, the mean keeps the unit of the data: adding beats per minute and dividing by a pure count leaves beats per minute. Second, the total uses every value, so a single unusual measurement enters the answer and can pull it away from the rest of the group.

For example

Five children weigh 18, 20, 22, 24 and 26 kg. The total is 110 kg, and 110 ÷ 5 = 22 kg, so the mean weight is 22 kg.

Pause and check yourself

A ward records four morning temperatures: 37.0, 38.0, 38.5 and 40.5 °C. Work out the mean and state its unit.

Show the answer

The total is 37.0 + 38.0 + 38.5 + 40.5 = 154.0 °C. There are four observations, so the mean is 154.0 ÷ 4 = 38.5 °C. The unit is degrees Celsius, the same unit as the observations themselves.

Read detailed explanations
Why the mean is a balance point, and how far one value moves it

Look again at the mean as a physical balance. Imagine the observations placed as equal weights along a ruler; the mean is the point where the ruler rests level, its centre of gravity. This picture explains the mechanism. Around the mean, the distances of the values below it exactly cancel the distances of the values above it: these signed distances, called deviations, always add to zero. That identity is not a coincidence but the defining property of the arithmetic mean, and it is why the mean responds to the size of a value and not merely to its presence. A single observation sitting far from the rest contributes a large deviation, and the balance point must shift to restore equilibrium. The size of that shift can be predicted without recalculating everything. When one new observation x joins a set of n values whose mean was m, the mean moves by exactly (x − m) ÷ (n + 1), because the total gains x while the count gains one. With n = 7, m = 78 and x = 140, the shift is (140 − 78) ÷ 8 = 7.75 beats per minute, exactly the movement seen in Tutorial 2. The formula also shows why larger samples resist a single rogue value: the same spike divided by a bigger denominator moves the mean less. The median has no such sensitivity, because it is a positional statistic rather than an algebraic sum; inserting a value only changes which positions are central, and the magnitude of an extreme observation can never enter the calculation. This is why a distribution contaminated by a few very large readings is summarised by the median, while the mean is reserved for data in which every observation fairly represents the group.

Which average is legal? Levels of measurement and the shape of the data

The choice of average is not a matter of taste; it is constrained by the level of measurement of the variable and by the shape of its distribution. Nominal data are labels with no order — blood group, diagnosis, ward, marital status. Nothing can be added and nothing can be ranked, so the mean and the median are both undefined and the mode is the only permissible average. Ordinal data can be ranked, but the gaps between ranks are not equal — pain scored 0 to 10, an Apgar score, a satisfaction rating. The median is legitimate here because it uses only order, while the mean pretends that the step from 1 to 2 equals the step from 9 to 10. Interval and ratio data carry equal units, so all three measures are available, and the deciding factor becomes symmetry. In a symmetric distribution the mean and the median coincide, and the mean is preferred because it uses every observation. In a skewed distribution they separate, and the direction of the skew is readable from which one is larger: a long tail of high values — hospital charges, length of stay, viral load — drags the mean upward, so the mean exceeds the median; a long tail of low values, such as age at death in a neonatal unit, pulls the mean below the median. The rule for clinical reporting follows directly: use the mean with a symmetric variable, the median with a skewed one, and the mode with a category; and whenever the mean and the median differ appreciably, say so, because that disagreement is itself a finding about the data.

Averaging grouped data: weighted means from a frequency table

Small sets can be listed one value at a time, but ward data are usually recorded in a frequency table, as in Section 4.3: a column of classes and a column showing how many patients fall into each. Listing sixty weights individually is unnecessary. When values repeat, or when they are grouped into intervals, the mean becomes a weighted mean: mean = Σ(f × x) ÷ Σf, where f is the frequency of a value and x is that value, or the midpoint of its class. The Greek capital letter Σ means “add up what follows”. Each value is multiplied by the number of times it occurs, those products are added, and the result is divided by the total number of observations. A worked example shows why the midpoint matters. Suppose 60 patients are grouped by weight as 40–49 kg with f = 5, 50–59 kg with f = 12, 60–69 kg with f = 20, 70–79 kg with f = 15 and 80–89 kg with f = 8. The midpoints are 44.5, 54.5, 64.5, 74.5 and 84.5 kg. The products are 222.5, 654, 1,290, 1,117.5 and 676, which total 3,960. Dividing by Σf = 60 gives a weighted mean of 66 kg, an approximation because every patient in a class is treated as if they sat at its midpoint. Two cautions follow. First, a grouped mean is always an estimate, and the true mean can be recovered only from the raw values. Second, the mode of grouped data is the class with the largest frequency, called the modal class — here 60–69 kg, with f = 20 — and not a single weight, because the individual weights are no longer visible. The same table yields the median through cumulative frequencies: find the class in which the running total first passes n ÷ 2 = 30, then interpolate within it.

Explore the diagram

Look at the whole drawing first. Then select one numbered part and connect it to the idea you just read.

Mean, median and mode on one dataset

A three-panel teaching plate built on one small set of pulse rates. The top panel states the three formulas in words and works all three measures on 68, 72, 72, 78, 80, 84 and 92 beats per minute. The middle panel adds one extreme reading of 140 and shows the ordered values with the two middle positions marked, so the mean moves from 78.0 to 85.8 while the median moves only from 78 to 79. The bottom panel is the decision guide: which measure suits which shape of data, and how to report it with its unit.

New to diagrams? Read this first
  1. Look at the whole drawing once without clicking anything. Ask yourself only: what kind of picture is this (a comparison, a route, a sequence, a table)?
  2. Press Play, or use the arrows, to move through the numbered parts in order. The drawing stays still; only the highlight moves.
  3. Click a number, or its name in the list below, to read what that part is and why it matters. Use “Hide labels” when you want to test your memory.
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01 / 5The mean: a total divided by a count

Mean, median and mode on one datasetA three-panel teaching plate built on one small set of pulse rates. The top panel states the three formulas in words and works all three measures on 68, 72, 72, 78, 80, 84 and 92 beats per minute. The middle panel adds one extreme reading of 140 and shows the ordered values with the two middle positions marked, so the mean moves from 78.0 to 85.8 while the median moves only from 78 to 79. The bottom panel is the decision guide: which measure suits which shape of data, and how to report it with its unit.Three measures, three questionsMean = total of all values ÷ nMedian = middle value of ordered dataMode = value occurring most oftenPulse rates (bpm): 68 72 72 78 80 84 92Total 546, n = 7 → mean = 546 ÷ 7 = 78Ordered: 4th value → median = 78 bpm72 occurs twice → mode = 72 bpmEvery measure carries the data’s unit.One extreme value added: 140 bpmOrdered data; the two middle values68727278808492140Mean 78.0 → 85.8 bpm (moves 7.8)Median 78 → 79 bpm (moves 1)n = 8 → positions 4 and 5: (78 + 80) ÷ 2 = 79Here the median changes less than the mean.Which measure fits the data?Symmetric data, no extremes → meanA few very large or small values → medianCategories (blood group, ward) → modeEvery value once → no modeTwo values tie for most often → two modesName the measure and give its unit.The mean uses every value; the median does not.

If the drawing is wider than this panel, scroll it sideways. Labels stay readable; the numbered explanations below show the same ideas.

The mean: a total divided by a count

The mean answers what each patient would receive if the total were shared equally. Here the seven pulse rates total 546 beats per minute and n = 7, so the mean is 546 ÷ 7 = 78 beats per minute. Because every observation enters the total, one extreme reading can change the answer a great deal; because the divisor is a pure count, the mean keeps the unit of the data.

Teaching plate built on one schematic dataset of pulse rates; the data points are evenly spaced for legibility and are not drawn to a numerical scale. The 140 beats per minute reading is an extreme value used to show the difference between the mean and the median, not a typical observation.

Key points to remember

  • Mean = total of every observation ÷ n, and the answer keeps the unit of the data.
  • Median = the value at position (n + 1) ÷ 2 of the ordered data; when n is even, take the mean of the values at n ÷ 2 and (n ÷ 2) + 1.
  • Mode = the value that occurs most frequently. There may be no mode, one mode, two modes or several.
  • The mean uses every observation, so one extreme value moves it a lot; the median uses position only, so it hardly moves.
  • Let the shape of the data choose the measure: mean for even data, median for uneven data, mode for categories — and always name the measure with its unit.
Go deeper

Worked example

Describing the pulse rates of five patients. Five patients on a ward have their pulse rates recorded as 72, 80, 76, 84 and 78 beats per minute. A colleague asks for the mean, the median and the mode before deciding whether the ward looks typical or whether one patient needs review.

  1. Order the data first, because the median cannot be read from an unordered list. Smallest to largest: 72, 76, 78, 80, 84 beats per minute.
  2. Add every observation for the mean: 72 + 80 + 76 + 84 + 78 = 390 beats per minute. Count the observations to get n = 5.
  3. Divide the total by n: 390 ÷ 5 = 78. The mean pulse rate is 78 beats per minute.
  4. Find the median position with the formula (n + 1) ÷ 2 = (5 + 1) ÷ 2 = 3. The third ordered value is 78, so the median is 78 beats per minute.
  5. Count repeats for the mode. Each of 72, 76, 78, 80 and 84 appears exactly once, so no value occurs more frequently than the others and the data have no mode.

The mean and the median agree at 78 beats per minute, which tells you the five readings are fairly even, and there is no mode because nothing repeats. Reporting “no mode” is a complete answer, not a failure: it is the correct description of data in which every value occurs the same number of times.

Tutorial 1 — mean, median and mode of seven pulse rates

Seven patients on a surgical ward have their pulse rates recorded as 84, 68, 92, 72, 78, 72 and 80 beats per minute. A nurse must summarise the ward in three numbers: one that shares the total fairly, one that describes the middle patient, and one that names the most common reading.

Rule used: Mean = total of every value ÷ n, where n is how many values there are; the answer keeps the unit of the data. Median = the value in position (n + 1) ÷ 2 of the ordered list when n is odd. Mode = the value that occurs most frequently.

  1. Put the seven readings in ascending order before doing anything else.

    Reveal this step

    68, 72, 72, 78, 80, 84 and 92 beats per minute. Ordering first prevents the commonest error, which is reading the middle of the list exactly as it was written.

  2. Add all seven values to find the total, and write down n.

    Reveal this step

    68 + 72 + 72 + 78 + 80 + 84 + 92 = 546 beats per minute, and n = 7 observations.

  3. The total is 546 beats per minute and n = 7. What is the mean, and what does the unit tell you?

    Reveal the answer

    Mean = 546 ÷ 7 = 78 beats per minute. Dividing a total measured in beats per minute by a pure count leaves the unit unchanged, so the mean is itself a pulse rate and not a number of patients.

  4. Find the median position with the formula, then read the value at that position.

    Reveal this step

    Position = (n + 1) ÷ 2 = (7 + 1) ÷ 2 = 4, so the median is the 4th ordered value, 78 beats per minute.

  5. The mean and the median are both 78 beats per minute. Does that prove the two measures always agree?

    Reveal the answer

    No. They agree here only because the seven readings are fairly even. A single extreme reading separates them sharply, and the next tutorial adds exactly such a value to this same set.

  6. Count how often each value appears and name the mode.

    Reveal this step

    68, 78, 80, 84 and 92 each appear once; 72 appears twice. The mode is 72 beats per minute, because it is the only value that repeats.

  7. A colleague reports “the average pulse is 72 beats per minute”. What has the colleague done?

    Reveal the answer

    The colleague has reported the mode and called it the average. The mean is 78 beats per minute, so the statement understates the ward by 6 beats per minute. Naming the measure used — mean, median or mode — prevents exactly this confusion.

Answer: Mean 78 beats per minute, median 78 beats per minute and mode 72 beats per minute; the three numbers together describe the ward.

Keep this habit: Order the data before finding the median, add every value before dividing for the mean, and count repeats for the mode. Always name the measure you report, because the same dataset gives three different and equally correct “averages”.

Tutorial 2 — one extreme value and what it does to each measure

A patient with severe tachycardia is admitted to the same ward, so the eight pulse rates are now 68, 72, 72, 78, 80, 84, 92 and 140 beats per minute. Before recalculating, the nurse wants to know which of the three measures will move and by how much.

Rule used: Mean = total of every value ÷ n. When n is even the median is the mean of the two middle values: let P1 = n ÷ 2 and P2 = P1 + 1, then median = (value at P1 + value at P2) ÷ 2. When a new value x joins n values whose mean was m, the mean moves by (x − m) ÷ (n + 1).

  1. Write the eight readings in ascending order and mark the two middle positions.

    Reveal this step

    68, 72, 72, 78, 80, 84, 92 and 140. With n = 8, the two middle positions are the 4th and 5th: 78 and 80 beats per minute.

  2. Before any calculation, predict which measure will move further: the mean or the median. Give your reason.

    Reveal the answer

    The mean, because the total absorbs the whole size of 140. The median asks only which values sit in the middle, so a value at the end of the list changes nothing except which two values are central.

  3. Add all eight values and divide by n to get the new mean.

    Reveal this step

    68 + 72 + 72 + 78 + 80 + 84 + 92 + 140 = 686, and 686 ÷ 8 = 85.75 ≈ 85.8 beats per minute.

  4. The mean was 78.0 and is now 85.75. Use the change rule (140 − 78) ÷ 8 to check the movement. Do the two routes agree?

    Reveal the answer

    (140 − 78) ÷ 8 = 62 ÷ 8 = 7.75, and 78.0 + 7.75 = 85.75. Both routes agree: one new value of 140 has raised the mean by 7.75 beats per minute.

  5. Now recompute the median from the two middle values.

    Reveal this step

    (78 + 80) ÷ 2 = 79 beats per minute. The median has moved by 1 beat per minute, from 78 to 79.

  6. The outlier moved the mean by 7.75 and the median by 1. Which number belongs in a ward report, and what would you say about the other?

    Reveal the answer

    Report the median, 79 beats per minute, as the typical pulse for the ward, and mention the mean of 85.8 with the note that one patient with severe tachycardia pulled it up. Reporting the mean alone would make the ward look far more tachycardic than it is.

Answer: After the 140 beats per minute reading is added, the mean rises from 78.0 to 85.8 beats per minute while the median rises only from 78 to 79.

Keep this habit: Use the median whenever one or two extreme observations could dominate the picture, and use (new value − old mean) ÷ (n + 1) to predict the mean’s movement before recalculating. A mean and a median that disagree are telling you the data are uneven.

Now you try

Attempt each one on paper first. Write the formula, substitute the numbers, then check the unit of your answer.

  1. Six patients’ ages are 24, 28, 28, 31, 35 and 42 years. Work out the mean, the median and the mode, and round the mean to one decimal place.

    Hint

    Order the ages, note that n = 6, and remember that the median of an even set is the mean of the values in positions 3 and 4.

    Show the worked answer

    Total = 24 + 28 + 28 + 31 + 35 + 42 = 188 years, and n = 6, so the mean is 188 ÷ 6 = 31.333… ≈ 31.3 years. The two middle ordered values are in positions 3 and 4: 28 and 31, so the median is (28 + 31) ÷ 2 = 29.5 years. 28 appears twice and every other age once, so the mode is 28 years. The mean sits above the median because the two oldest patients stretch the upper end.

  2. A clinic records seven waiting times in minutes: 12, 14, 15, 15, 17, 18 and 90. Find the mean and the median, and say which one describes a typical wait.

    Hint

    Add all seven values, then ask which single number the mean is using heavily. The median position for n = 7 is (n + 1) ÷ 2 = 4.

    Show the worked answer

    Total = 12 + 14 + 15 + 15 + 17 + 18 + 90 = 181 minutes, and 181 ÷ 7 = 25.857… ≈ 25.9 minutes. The median is the 4th ordered value, 15 minutes, and the mode is also 15 because it appears twice. The median describes a typical wait: six of the seven patients waited 18 minutes or less, while the single 90-minute wait lifts the mean to 25.9. Report the median, and give the mean as evidence of the long wait.

  3. Eight patients’ pain scores out of 10 are 1, 2, 2, 3, 3, 4, 4 and 8. Which measures of central tendency may legitimately be reported for these scores, and what is the value of each?

    Hint

    A pain score is ordinal: its values can be ranked but the gaps between them are not known to be equal. Ask which measures use only the order, and which one assumes equal steps.

    Show the worked answer

    The median and the mode are legitimate, but the mean is not. The list is already ordered and n = 8, so the median averages the fourth and fifth values: (3 + 3) ÷ 2 = 3 points. The values 2, 3 and 4 each appear twice, so they tie for most frequent and the data have three modes; 1 and 8 appear once. The mean would be 27 ÷ 8 = 3.375, but reporting 3.4 “points” would claim that the step from 1 to 2 equals the step from 9 to 10, which a 0–10 pain scale does not promise — the same reason section 4.2 refuses a mean for an ordinal scale. Median and modes are reported as values on the scale.

Common misconception

Beginners often treat the three measures as interchangeable names for “the average”, so they compute whichever is easiest and label the result average. That is how a ward with a mean pulse of 78 and a mode of 72 can be described in two ways that look contradictory although both are correct. The measures are not synonyms: each answers a different question, and each responds to different features of the data. A second common mistake is taking the median of an unordered list, or forgetting to halve the two middle values when n is even. Sort first, then read the position.

Check your reasoning

A hospital reports that the mean length of stay on a ward is 9 days, yet the ward manager says most patients go home after 4 days. Using the mean and the median, explain how both statements can be true at the same time.

Reveal an answer

Both can be true because the two measures answer different questions. Length of stay almost always contains a few very long admissions — complex surgery, complications, delays in social care — and the mean adds every one of those days into the total, so a small number of long stays lifts it well above what most patients experience. The median is decided by position: if most patients stay about 4 days, the middle value of the ordered list is 4 days, however long the longest admission lasted. The manager is describing the typical patient (median) and the hospital is describing the average share of bed-days (mean). Best practice is to report both, each with its unit and its name.

Where this comes from

  • BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Unit Four “Statistical Measures”, lines 758–822: the measures of position and central tendency; the arithmetic mean with the 4, 6, 8, 10, 12 and pulse-rate examples; the median with its odd- and even-number examples; the mode with the 2, 3, 3, 4, 5, 3, 6 example and the list of possible mode counts; and the closing comparison of the mean, median and mode in medical data. The grouped-weight table reused in the deeper topic is at lines 760–766.
  • Beyond the supplied notes: formulas and applications the notes state only by example · the even-number median written as a formula (P1 = n ÷ 2, P2 = P1 + 1), the balance-point mechanism with the change rule (x − m) ÷ (n + 1), the constraint that the level of measurement decides which average is legal, and the weighted mean Σ(f × x) ÷ Σf for grouped data with the modal class. The notes give odd and even examples and a frequency table but do not state these as rules.

Lesson progress is saved on this device for the current learner. Practice answers stay separate and are not stored.

About this course and its sources

Bilingual lessons organised from the supplied BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (Mr. Ebangan Prosper Achuanoh). Diagrams are original educational schematics; sections that go beyond the supplied notes say so in their sources. Mathematics, clinical and French-language review are pending.