Comparing distributions: averages and spread
An average alone can hide almost everything interesting in a data set. Comparing two groups properly means describing both a typical value and how spread out the values are.
Part of your national curriculum
- Statistics: Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread
Lesson overview
What you'll learn in this lesson
Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread
Key learning points
- • Choosing an average
- • Measuring spread
- • Mean from grouped data
- • Writing a comparison
This lesson at a glance
- 30 minutes
- 21 parts to scroll through
- 4 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 21
Visual introductionPicture this
Comparing distributions: averages and spread
An average alone can hide almost everything interesting in a data set. Comparing two groups properly means describing both a typical value and how spread out the values are.
In a nutshell
Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread
Learning cycle
Part 2 of 21
Learning cycle 1 of 2
Part 1 · Choosing an average
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 3 of 21
Learn
Choosing an average
The mean uses every value but is dragged by extremes. The median is the middle value once ordered and resists outliers, which is why house prices and incomes are reported as medians. The mode is the most common value and is the only average available for categorical data such as favourite subject.
Reset break
Part 4 of 21
Pause
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Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Explore the idea
Part 5 of 21
Learn
Measuring spread
The range is the largest value minus the smallest, and is easy but sensitive to a single outlier. The interquartile range, the upper quartile minus the lower quartile, describes the middle half of the data and is far more stable when a data set has an unusual value.
Quick check
Part 6 of 21
Quick check
Part 7 of 21
Reset break
Part 8 of 21
Pause
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Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Learning cycle
Part 9 of 21
Learning cycle 2 of 2
Part 2 · Mean from grouped data
A short piece of teaching, then a check to make sure it has landed.
Explore the idea
Part 10 of 21
Learn
Mean from grouped data
With grouped data the exact values are unknown, so use midpoints: for 10 values in 0 < t ≤ 10, 30 in 10 < t ≤ 20 and 10 in 20 < t ≤ 30, the estimated mean is (10×5 + 30×15 + 10×25) ÷ 50 = 15. The answer must be described as an estimate, because the midpoint assumption may not hold.
Explore the idea
Part 11 of 21
Learn
Writing a comparison
A comparison needs both measures and a context: 'Class A has a higher median score (32 against 27), so typically performed better, but a larger interquartile range (14 against 6), so its results were less consistent.' Quoting numbers without interpretation, or interpretation without numbers, is a half answer.
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Part 12 of 21
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Quick check
Part 13 of 21
Quick check
Part 14 of 21
Explore the idea
Part 15 of 21
Worked example
Worked answer: compare two classes, A (median 32, IQR 14) and B (median 27, IQR 6) (4 marks)
Class A has the higher median, 32 against 27 (1), so a typical student in A scored about five marks more (1). Class A also has the larger interquartile range, 14 against 6 (1), so its middle half of results is far more spread out, meaning B was more consistent even though it was typically lower-scoring (1).
Reset break
Part 16 of 21
Pause
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Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Challenge round
Part 17 of 21
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 18 of 21
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
The range is the largest value minus the smallest, and is easy but sensitive to a ____ outlier.
Challenge round
Part 19 of 21
Game · Recall cards
Increase 240 by 15%.
Card 1 of 4
Mastery quiz
Part 20 of 21
Marked quiz
End of lesson quiz: Comparing distributions: averages and spread
4 questions, marked with the reasoning shown. No timer.
1. Increase 240 by 15%.
2. A count drops from 200 to 150. The percentage decrease is…
3. Which multiplier applies a 7% decrease?
4. Which average is least affected by one huge value?
Lesson round-up
Part 21 of 21
Lesson round-up
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Describe, interpret and compare observed distributions of a single variable through appropriate measures of central tendency and spread