Identifying distribution shape
A distribution is described by the way its values are spread around the main cluster. Check the long tail, because its direction names a skew
\[\text{positive skew}\ \Rightarrow\ \text{long tail toward larger values}\]
- Find the peak and tails Look for the main concentration of values and the thin side that stretches furthest from it \(\displaystyle \text{heights }1,2,6,3,2,1,1\)
- Recognise symmetry A distribution is approximately symmetric when the two sides mirror each other about the middle \(\displaystyle 1,3,5,3,1\quad\text{is approximately symmetric}\)
- Name positive skew A positive skew has its long tail toward the larger values \(\displaystyle \text{tail}\rightarrow\text{larger values}\quad\Rightarrow\quad\text{positive skew}\)
- Name negative skew A negative skew has its long tail toward the smaller values \(\displaystyle \text{tail}\rightarrow\text{smaller values}\quad\Rightarrow\quad\text{negative skew}\)
Worked example
A histogram of course-completion times has seven bins with the following heights. Describe the shape of the distribution
\[\text{bin positions: }1,2,3,4,5,6,7\quad\text{heights: }1,2,6,3,2,1,1\]
- \(\displaystyle \text{largest height}=6\quad\text{at bin }3\)The main cluster peaks at bin 3.
- \(\displaystyle 7-3=4\quad\text{bins to the right};\quad 3-1=2\quad\text{bins to the left}\)The values extend further on the larger-value side of the peak
- \(\displaystyle \text{long tail toward larger values}\quad\Rightarrow\quad\text{positively skewed}\)The tail points toward larger completion times, so the distribution is positively skewed
Common mistakes
Naming skew from the peak
✕\(\displaystyle \text{peak toward larger values}\Rightarrow\text{positive skew}\)✓\(\displaystyle \text{tail toward smaller values}\Rightarrow\text{negative skew}\)
The skew is named by the direction of the long tail, not by the side where the peak is locatedReading the median as the box midpoint
✕\(\displaystyle \text{median}=\frac{Q_1+Q_3}{2}\)✓\(\displaystyle \text{median} = \text{the marked line between }Q_1\text{ and }Q_3\)
In a boxplot the median is read from its marked line, which need not be at the midpoint of the boxUsing the mean for a skewed distribution
✕\(\displaystyle \text{skewed distribution}\quad\Rightarrow\quad\text{mean is preferred}\)✓\(\displaystyle \text{skewed distribution}\quad\Rightarrow\quad\text{median is preferred}\)
The mean is pulled toward a long tail, so the median is usually the better measure of centreDifficulty levels
introA clear histogram has a single peak and an unmistakable shape to classify
standardA dot plot or smaller histogram has a shorter tail and may include a centre or spread comparison
challengeA boxplot, a symmetric body with an outlier or a preferred-centre follow-on requires a less direct shape reading
examA choice identifies the shape of a rendered display
Questions about this topic
How are positive and negative skew named?+
Positive skew has a long tail toward larger values, while negative skew has a long tail toward smaller values
What does approximately symmetric mean?+
The distribution has a similar pattern on both sides of its middle, allowing for small differences in the frequencies
Why is the median often preferred for skewed data?+
The median is less affected by the long tail than the mean, so it gives a more representative centre for a skewed distribution
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 11: Binomial distribution
In this chapter
Bernoulli trials and the Bernoulli distribution11A Bernoulli trials and the Bernoulli distribution: Finite events, probability assignments and complementsSolve a quadratic exactly with the quadratic formula11A Bernoulli trials and the Bernoulli distribution: Round to decimal places and significant figures11B Binomial distribution and parameters: Binomial conditions, the distribution table and graph shapeMean, variance, standard deviation and parameter recovery of a binomial variableExact probability of repeated independent successesEvaluate factorial and combination expressionsSolve a linear equation11B Binomial distribution and parameters: Round to decimal places and significant figuresCumulative, at-least-one and conditional probabilities of a binomial variableSmallest trial count for an at-least-one threshold11C Binomial distribution events: Finite events, probability assignments and complementsConditional probability from exact representationsSolve and interpret a linear inequation11D Binomial shape and specified order: Binomial conditions, the distribution table and graph shapeProbabilities of sequences of independent eventsDescribe the shape of a distribution