Lesson 5 of 9 / Practical and data skills
Describe both the centre and the spread
Can two samples have the same mean but tell different stories?
I can calculate a mean and sample standard deviation, and explain what each describes.
About 6 min
Make a guess. You are not marked.
Two samples both have a mean height of 20 cm. Must they look alike?
Show the answer
Their spreads may be very different.
Standard deviation measures spread around the mean. One sample could be tightly clustered, the other widely spread.
The key idea
The mean describes the centre. Standard deviation describes the spread of observations around that mean, in the same units as the measurements.
Work with the evidence
Same mean, different spread
Values: 3, 4, 5, 6, 7 cm. Mean = 5 cm in both samples. Sample standard deviation = 1.58 cm. Wider spacing increases spread without moving the mean.
Explanation
Calculate a sample mean by adding all values and dividing by the number of observations. A mean is a summary, not a replacement for the raw data. Two samples can have the same mean while one contains much more widely spread observations.
For the sample standard deviation supplied in the syllabus, subtract the mean from each observation, square each difference, add the squared differences, divide by n - 1, then take the square root. Squaring prevents positive and negative deviations from cancelling. Taking the square root returns the result to the measurement unit.
For 3, 4, 5, 6 and 7 cm, the mean is 5 cm. The squared deviations are 4, 1, 0, 1 and 4 cm2, summing to 10 cm2. Divide by 4 and take the square root: s is about 1.58 cm. These are sample observations, so follow the supplied sample formula.
A larger standard deviation means greater spread, not a larger mean, greater accuracy or proof of an experimental mistake. Spread may arise from biological differences and measurement variation. An instrument with a consistent offset can give tightly grouped but inaccurate readings.
A normal distribution is symmetric and bell-shaped, with many observations near its centre and fewer far from it. Some statistical methods rely on an approximately normal underlying distribution. A small sample does not establish that assumption merely because its mean and standard deviation can be calculated. Error bars labelled as standard deviations describe spread; overlap alone is not a substitute for the specified significance test.
Step by step
- 1
Find the mean
Add the observations and divide by n.
- 2
Measure squared deviations
Subtract the mean, square each difference and add them.
- 3
Complete and interpret
Divide by n - 1, take the square root and describe the spread in the original units.
Worked example
Work through the evidence
Compare samples 3, 4, 5, 6, 7 cm and 1, 3, 5, 7, 9 cm.
One way to explain it
Both means are 5 cm. The first sample has s = square root of (10/4), about 1.58 cm. The second has s = square root of (40/4), about 3.16 cm. The second sample has greater spread despite an identical mean.
Why this answer works
- State the shared centre.
- Calculate spread from deviations, not the range alone.
- Interpret the larger standard deviation without calling the data wrong.
Watch out for this
A student says: "A small standard deviation proves the measurements are accurate." What is wrong with this?
Show the answer
Values can cluster tightly around a biased value. Standard deviation describes spread, while accuracy concerns closeness to the true value.
Check your understanding
What happens to the standard deviation if the same 2 cm calibration error is added to every observation?
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It remains unchanged, while the mean rises by 2 cm.
A shared offset changes the centre but not the deviations around it.
Exam-style questions
Check your understanding
Which statement about standard deviation is NOT correct?
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It is always smaller than the mean
Spread can exceed the mean when values vary widely, so this is not a rule.
Quick recall
Can two samples have the same mean but tell different stories?