NCEA Level 2Comparing distributionsAS 91267

Comparing distributions: shape, centre and spread

What this lesson covers

  • A distribution is described by its shape, centre and spread, each with numbers.
  • Whether a normal model is appropriate depends on the shape of the data.
Key idea

Describe with three words, and numbers

  • Shape, centre and spread.
  • Every feature needs a number.
  • Every comparison needs both groups.
Worked example

Commute times in two towns

  • Town A: a single hump in the middle.
  • Town B: piled up on the left, a long tail right.
  • Same scale, 48 workers each.
Commute times in two towns
Worked example

Where the mean and median sit

  • Town A: mean and median together, about 35.
  • Town B: median about 25, mean 27.7.
  • The tail drags B's mean to the right.
Where the mean and median sit
Worked example

The means from grouped data

  • Use each class midpoint.
  • Σ count × midpoint, ÷ 48.
  1. midpoints 5, 15, 25, …, 65class middles
  2. A: 1×5 + 4×15 + 10×25 + 16×35
  3. + 11×45 + 5×55 + 1×65
  4. = 1710
  5. mean = 1710 ÷ 48 = 35.6 min
  6. B: 4×5 + 14×15 + 12×25 + 8×35
  7. + 5×45 + 3×55 + 2×65
  8. = 1330
  9. mean = 1330 ÷ 48 = 27.7 min
Worked example

The medians from grouped data

  • The middle: 24th, 25th.
  • Use cumulative counts.
  1. n = 48: 24th and 25th values
  2. A: cumulative 1, 5, 15, 31
  3. both in the 30–40 class15 < 24, 31 ≥ 25
  4. A: median ≈ 35 minclass middle
  5. B: cumulative 4, 18, 30
  6. both in the 20–30 class18 < 24, 30 ≥ 25
  7. B: median ≈ 25 minclass middle
  8. B: mean 27.7 > median 25right skew
Worked example

The summary statistics

  • Estimated from the grouped data.
  • Means, medians and standard deviations.
  • These are the numbers for your sentences.
Town ATown B
Mean35.6 min27.7 min
Medianabout 35 minabout 25 min
Standard deviation12.6 min15.5 min
Shapesymmetricright-skewed
Key idea

Shape: symmetric or skewed

  • The tail names the skew: right tail, right skew.
  • Skew pulls the mean towards the tail.
  • So mean > median signals right skew.
ShapeLooks likeHere
Symmetricmean ≈ medianTown A
Right-skewedright tail, mean > medianTown B
Left-skewedleft tail, mean < median
Key idea

Which centre to use

  • Symmetric data: mean or median, they agree.
  • Skewed data: the median.
  • A few long commutes do not move the median.
Key idea

Spread, with a number

  • Town B: standard deviation 15.5 minutes.
  • Town A: 12.6 minutes.
  • B's commutes are more variable.
Worked example

A full comparison

  • One sentence per feature.
  • Both towns in every sentence.
  • A number in every sentence.
FeatureComparison
ShapeA symmetric; B skewed right
CentreA median about 10 min higher
SpreadB more variable: sd 15.5 vs 12.6
UnusualB has a few very long commutes
Key idea

Is a normal model appropriate?

  • A normal model is symmetric with one peak.
  • Town A fits that; Town B does not.
  • Saying why is an Excellence point.
ShapeSo
Town Asymmetric, one peaknormal model fits
Town Bskewednormal model does not fit
Key idea

Where the marks are

  • Achieved: one feature compared.
  • Merit: two features, with numbers.
  • Excellence: all three, plus the model's fit.
Summary

What to take away

  • Shape, centre, spread: both groups, with numbers.
Shapesymmetric or skewed, and which way
Centremedian for skewed data
Spreadsd or IQR, with numbers
Skewpulls the mean towards the tail
Compareevery feature, both groups, numbers
Normal modelonly for symmetric, one-peaked data
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Use numbers from the summary.
  • Name both classes every time.
Question
Xmean 64, median 65, sd 8, symmetric
Ymean 58, median 54, sd 14, right-skewed
1Compare the centres, with numbers.
2Compare the spreads, with numbers.
3Which class suits a normal model? Why?
4Why is Y's mean higher than its median?
Show answer

1. X is higher: median 65 against 54, 11 marks more. 2. Y is more spread out: standard deviation 14 against 8. 3. X: it is symmetric (mean close to median). Y is skewed, so normal model does not fit. 4. The right skew: a few high marks pull the mean up, while the median is not affected by them.

Worked example

The answers

  1. 1. X median 65, Y median 54
  2. 65 − 54 = 11: X is 11 higher
  3. 2. sd: Y 14, X 8
  4. Y is more spread out
  5. 3. X: mean 64, median 65close: symmetric
  6. X suits a normal model
  7. Y is skewed, so it does not
  8. 4. Y: mean 58 > median 54right skew
  9. high marks pull the mean up

Is your child on track?

Take the free placement test