Comparing Data Sets and Shape
Theory
The shape of a distribution shows how the data are spread out, and comparing two sets means comparing their centre, spread and shape. This Year 9 maths page shows how to describe data as symmetrical, positively skewed, negatively skewed, uniform or bimodal, use the mean and median to tell the skew without a picture, and compare two data sets in words and numbers. Curriculum: VC2M9ST03 / AC9M9ST03.
The shape of a distribution is the outline its histogram or dot plot makes. Look for where most of the data sit and which way the tail (the thin end) runs.
A symmetrical distribution has two halves that match when folded down the middle.
A positively skewed distribution has most scores at the low end and a long tail running to the right, towards the larger numbers. A negatively skewed distribution has most scores at the high end and a tail running to the left. The skew is named after the tail, not the hump.
Uniform data have every column the same height, so there is no mode. Unimodal data have one peak and bimodal data have two.
To compare two data sets, look at three things. Centre says which set is higher overall: compare the two means, or the two medians. Spread says which set is more consistent: compare the ranges, and a smaller range means the scores are closer together. Shape says which way each set leans.
Two sets can have the same mean and be nothing alike. Displays drawn on the same scale, such as two dot plots one above the other or a back-to-back stem-and-leaf plot, let you compare all three by eye.
The tail drags the mean towards it, but the median barely moves. So comparing them tells you the shape:
| Shape | Where most scores sit | Mean and median |
|---|---|---|
| symmetrical | evenly about the centre | \(\text{mean} = \text{median}\) |
| positively skewed | at the low end, tail to the right | \(\text{mean} > \text{median}\) |
| negatively skewed | at the high end, tail to the left | \(\text{mean} < \text{median}\) |
| uniform | spread evenly, no peak | no mode |
| Compare | Using | It tells you |
|---|---|---|
| centre | the two means, or the two medians | which set is higher overall |
| spread | the two ranges | which set is more consistent |
| shape | each set's mean against its own median | which way each set leans |
How to describe and compare distributions
- Draw or read the display. From raw data, sort and count each score, then make a dot plot.
- Name the shape: tail right is positive, tail left is negative, matching halves is symmetrical. Count the peaks: none (uniform), one (unimodal) or two (bimodal).
- Without a picture, compare each set’s mean with its median.
- Centre: find the same measure for both sets and say which is higher, and by how much.
- Spread: quote both ranges and say which set is more consistent.
- Conclude in context with both parts: “Set \(B\) scores higher on average, but set \(A\) is more consistent.”
Find where the bulk of the data sits, then which way the tail runs.
| \(\text{tall bars}\) | \(:\) | \(\text{at the HIGH scores}\) |
| \(\text{tail}\) | \(:\) | \(\text{runs LEFT}\) |
A tail to the left is a negative skew. The distribution is negatively skewed.
Compare each set’s own mean with its own median.
| \(E\) | \(:\) | \(30 = 30\) |
| \(F\) | \(:\) | \(41 > 34\) |
\(E\)’s mean equals its median, so \(E\) is symmetrical. \(F\)’s mean is above its median, so a tail of high values has dragged it up: \(F\) is positively skewed.
The medians, \(30\) and \(34\), are the fairer comparison of centre here, because \(F\)’s mean is pulled by its tail.
Compare the centres first, then the spreads.
| \(\text{mean } X\) | \(=\) | \(\dfrac{35}{5} = 7\) |
| \(\text{mean } Y\) | \(=\) | \(\dfrac{35}{5} = 7\) |
| \(\text{range } X\) | \(=\) | \(9 - 3 = 6\) |
| \(\text{range } Y\) | \(=\) | \(8 - 6 = 2\) |
The means are equal, so the spread decides. \(Y\) is far more consistent, and \(Y\)’s lowest score is \(6\), while \(X\) once scored \(3\). The coach should pick archer \(Y\).
Compare like with like: median with median, range with range.
| \(\text{median } C\) | \(=\) | \(26\) |
| \(\text{median } D\) | \(=\) | \(31\) |
| \(\text{range } C\) | \(=\) | \(33 - 21 = 12\) |
| \(\text{range } D\) | \(=\) | \(40 - 19 = 21\) |
\(D\)’s median is \(5\) higher, so \(D\) scores higher. But \(D\)’s range is \(9\) larger, so \(C\) is the more consistent set. A full answer gives both parts.
Common pitfalls
Frequently asked questions
What does positively skewed mean?
Most of the scores are at the low end and a long tail runs to the right, towards the larger numbers. The mean is greater than the median.
How do you tell if data are positively or negatively skewed?
Look at which way the tail runs. A tail to the right is positive and a tail to the left is negative. Without a picture, compare the mean and median: mean above median is positive, mean below median is negative.
What is the difference between uniform and bimodal?
Uniform data have every column the same height, so there is no peak. Bimodal data have two separate peaks.
How do you compare two sets of data?
Compare their centres using the means or medians, their spreads using the ranges, and their shapes using each set's mean and median. Then write a conclusion that gives both the centre and the spread, in context.
Can two data sets have the same mean but be different?
Yes. \(7\), \(8\), \(8\), \(9\), \(9\), \(9\), \(10\), \(10\), \(11\) and \(3\), \(5\), \(7\), \(8\), \(9\), \(10\), \(11\), \(13\), \(15\) both have a mean of \(9\), but the second set has three times the range.