Mean, median and normal distribution
Mean, Median, Mode, Standard Deviation and the Normal Distribution
How to find mean, median and mode, which to use for skewed data, and how SD, IQR, centiles and standard error differ, with worked FCPS examples. Covers the normal distribution (68-95-99.7), reference ranges, Z scores, skew, box plots and ogives.
One of the most tested topics in past papers
Measures of central tendency
| Measure | How to get it | Best for |
|---|---|---|
| Mean | Sum of values ÷ number of values | Symmetrical (normal) numerical data; uses every value but is pulled by outliers |
| Median | Middle value after ordering; the 50th centile | Skewed data, outliers, ordinal data (hospital stay, income, survival) |
| Mode | Most frequent value | Nominal data; can be bimodal (Hodgkin lymphoma age peaks) |
| Geometric mean | nth root of the product (mean of logs) | Log-normal data: antibody titres, bacterial counts |
- Nominal data allow only the mode; ordinal data the median (and mode); interval and ratio data all three.
- Always sort the data first when finding a median. Its position is (n + 1) ÷ 2. With an even number of values, average the two middle values.
Worked median and mean examples
- 20, 20, 25, 30, 35, 40, 25, 30: sorted = 20, 20, 25, 25, 30, 30, 35, 40. n = 8, position 4.5, so median = (25 + 30) ÷ 2 = 27.5. Mean = 225 ÷ 8 = 28.1; there are three modes (20, 25, 30), each appearing twice.
- 141, 141, 143, 145, 146, 155: n = 6, median = (143 + 145) ÷ 2 = 144. Mean = 871 ÷ 6 = 145.2; mode = 141. Do not confuse the median with the mean.
- Systolic pressures 110, 120, 130, 140, 250: mean = 150, pulled up by the single value of 250; median = 130 best describes the typical patient.
- 2, 3, 3, 4, 13: mean 5, median 3, mode 3: the outlier creates positive skew.
Exam trap Unsorted lists are a deliberate trap: put the values in order before picking the middle.
Measures of dispersion
| Measure | Definition | Pairs with |
|---|---|---|
| Range | Maximum − minimum; distorted by extremes | Median (simple description) |
| Interquartile range (IQR) | Q3 − Q1 (25th to 75th centile): the middle 50%; resistant to outliers | Median |
| Variance | Σ(x − mean)² ÷ (n − 1) | Calculations |
| Standard deviation (SD) | √variance; same units as the data | Mean |
| Coefficient of variation | SD ÷ mean × 100% | Comparing spread of measures with different units or means (lab precision) |
- Centiles divide ordered data into 100 parts. A child on the 50th centile for weight is at the median: half of children of that age and sex weigh less and half weigh more. Growth charts usually mark the 2nd or 3rd and 97th or 98th centiles as limits of normal.
- SD versus standard error: SD describes how individual values vary; the standard error of the mean (SE = SD ÷ √n) describes how precisely the sample mean estimates the population mean. Because SE depends on √n, halving the SE needs four times the sample size.
Box plots and ogives
- In a box-and-whisker plot the box spans the IQR (Q1 to Q3) and the line inside it is the median; whiskers reach the range (or 1.5 × IQR) and dots mark outliers. A median nearer one end of the box, or a longer whisker on one side, shows skew.
- An ogive is a cumulative frequency curve. Read across from 50% of the total to find the median, from 25% and 75% for the quartiles, and so on for any centile.
The normal (Gaussian) distribution
- Symmetrical and bell-shaped; mean = median = mode; fully described by its mean and SD; total area = 1.
| Range | Share of values |
|---|---|
| Mean ± 1 SD | About 68% |
| Mean ± 1.96 SD (about 2 SD) | 95% |
| Mean ± 2.58 SD | 99% |
| Mean ± 3 SD | 99.7% |
- Lab reference ranges are usually mean ± 2 SD, the central 95% of healthy people, so 5% of healthy people fall outside (2.5% at each end) by definition.
- Example: haemoglobin mean 140 g/L, SD 10 g/L: 95% of healthy people lie between 120 and 160 g/L; about 68% between 130 and 150.
- Z score (standard normal deviate) = (value − mean) ÷ SD; it tells how many SDs a value lies from the mean. WHO growth standards use Z scores: weight-for-height below −3 Z is severe wasting.
- Parametric tests (t-test, ANOVA, Pearson correlation) assume normality; check with a histogram, Q–Q plot or Shapiro–Wilk test, or by seeing that mean and median are close.
Remember 68, 95, 99.7: one, two and three standard deviations either side of the mean.
Skewed distributions
| Skew | Long tail | Order of averages | Examples |
|---|---|---|---|
| Positive (right) | To the right (high values) | Mean > median > mode | Income, hospital stay, triglycerides, incubation periods, number of children, viral load |
| Negative (left) | To the left (low values) | Mean < median < mode | Age at death in rich countries, gestational age at birth, Apgar scores |
- The mean is dragged toward the tail; the median stays with the bulk of the data. The mode sits at the peak.
- Report skewed data as median and IQR, use non-parametric tests, or log-transform right-skewed data.
- Kurtosis describes peakedness: leptokurtic (tall, thin) or platykurtic (flat).
Other distributions
- Binomial: number of successes in n trials with fixed probability (boys in families of four).
- Poisson: counts of rare events in fixed time or space; mean equals variance (cases per day, radioactive decay).
- t distribution: like the normal but with fatter tails for small samples; approaches normal as degrees of freedom rise.
- Chi-squared and F distributions underlie the tests of the same names.
How FCPS asks about this
- Median of 20, 20, 25, 30, 35, 40, 25, 30: sort first; 27.5.
- Typical value for 110, 120, 130, 140, 250 mmHg: median, 130 (the mean is distorted by 250).
- Child on the 50th centile: half weigh less, half more.
- What the box in a box plot shows: the IQR, with a line at the median.
- Curve used to read median and percentiles: ogive (cumulative frequency curve).
- Halving the standard error: increase sample size fourfold (SE = SD ÷ √n).
- Order of averages in positive skew: mean > median > mode; 95% of a normal distribution lies within 1.96 SD.
Test yourself on this topic
5 of the 26 questions in our bank on this topic. Choose an answer to see why every option is right or wrong.
A child's weight is on the 50th centile. What does this mean?
Answer: Half weigh less, half more
A percentile (centile) is the value below which a given percentage of observations fall. The 50th centile is the median. The 25th and 75th centiles are the lower and upper quartiles. On growth charts, weights between the 2nd and 98th centiles are usually considered normal; crossing two centile lines downward suggests faltering growth. The 2nd and 98th centiles correspond roughly to ±2 SD.
Centiles
- 50th centile = median
- 25th and 75th = quartiles
- Normal growth: 2nd–98th (≈ ±2 SD)
- Crossing two centile lines: faltering
Remember The 50th centile is right in the middle of the crowd.
Why the other options are wrong
- Weighs 50% of expected A centile is a rank, not a fraction of expected weight.
- Is 50% overweight Not the meaning.
- Is 2 SD below the mean That is about the 2nd centile.
- Is in the lowest 5% That is below the 5th centile.
A cumulative frequency curve is used to read off the median and percentiles. What is this curve called?
Answer: Ogive
An ogive plots cumulative frequency (running total) against the upper limit of each class interval, giving an S-shaped curve. Reading across from the 50% point gives the median; 25% and 75% give the quartiles, and other percentiles can be read similarly (used in growth charts). A frequency polygon joins the mid-points of histogram bars and shows the shape of a distribution.
Ogive
- Cumulative frequency curve, S-shaped
- Read median at 50%
- Read quartiles at 25% and 75%
Remember Ogive adds as it goes.
Why the other options are wrong
- Frequency polygon Shape of the distribution, not cumulative.
- Histogram Bars of frequency.
- Normal curve A theoretical bell-shaped distribution.
- Scatter diagram Two variables.
Fasting glucose in a population is normally distributed with mean 5.0 mmol/L and SD 0.5. What percentage of people have values above 6.0 mmol/L?
Answer: About 2.5%
6.0 is 2 SD above the mean. About 95% of values lie within ±2 SD (more precisely 1.96), leaving 5% outside, split equally: 2.5% above and 2.5% below. 1 SD covers 68% (16% above), 3 SD covers 99.7% (0.15% above). Reference ranges are usually the central 95%.
Normal curve tails
- ±1 SD: 68% (16% each tail)
- ±2 SD: 95% (2.5% each tail)
- ±3 SD: 99.7%
Remember Two SDs out leaves 2.5% on each side.
Why the other options are wrong
- About 5% Both tails together.
- About 16% Above 1 SD.
- About 32% Outside 1 SD both sides.
- About 0.15% Above 3 SD.
A study of 400 term babies in Multan reports a mean birth weight of 2.9 kg with a 95% confidence interval of 2.85 to 2.95 kg. What does this confidence interval mean?
Answer: The range likely to contain the true mean
A 95% confidence interval is a range calculated from the sample within which the true population mean is expected to lie; if the study were repeated many times, 95% of such intervals would contain the true mean. It reflects the precision of the estimate, is based on the standard error and narrows as sample size grows. It does not describe the spread of individual birth weights, which is given by the standard deviation.
95% confidence interval
- Likely range of the true population value
- Based on the standard error
- Narrows with larger samples
- Not the spread of individuals
Remember CI is about the mean, SD is about the people.
Exam trap A narrow confidence interval shows precision, not that the result is free of bias.
Why the other options are wrong
- The range of 95% of birth weights That is described by mean ± 1.96 SD of individual values.
- The range of normal birth weights A clinical reference range, not this.
- The probability the study is correct Not what a confidence interval means.
- The weights of 95% of the sample Describes spread of individuals, not precision of the mean.
A researcher wants to halve the standard error of a mean. By how much must the sample size be increased?
Answer: Four times
The standard error of the mean equals the standard deviation divided by the square root of the sample size (SE = SD/√n). Because of the square root, halving the standard error needs the sample size to be multiplied by four. Doubling the sample only reduces the SE by about 30%. This explains why big gains in precision become expensive.
Standard error
- SE = SD ÷ √n
- Halve SE: 4 × n
- Double n: SE ÷ 1.41
- Larger samples = narrower CIs
Remember Square root: four times the people for half the error.
Exam trap SD describes the data; SE describes the precision of the mean.
Why the other options are wrong
- Two times Doubling n reduces SE by only about 30%.
- Half as large A smaller sample increases SE.
- Eight times Eight times would reduce SE by more than half.
- Ten times Ten times reduces SE to about a third.
Where this sits in the CPSP syllabus
- Paper 1: Biostatistics, Central tendency and dispersion; normal distribution and skew