Statistics · Topic 4 of 13
Distributions
Theory
When working with continuous numerical data, plotting a histogram allows you to see the distribution (the overall shape and spread) of the data. Identifying the correct distribution is vital because it dictates which descriptive statistics you are allowed to use.
There are four main types of distributions you need to recognise:
1. Normal Distribution (Symmetrical)
- The histogram has a classic, symmetrical "bell shape".
- Most of the data clusters in the middle, with fewer values at the extreme high and low ends.
- Appropriate Measures: Because the data is symmetrical, you must use the mean for the measure of location and the standard deviation for the measure of spread.
2. Positively Skewed (Skewed to the Right)
- The bulk of the data is bunched up on the left side of the histogram, with a long "tail" stretching out to the right.
- This happens when there are a few unusually high outliers pulling the data upwards.
- Appropriate Measures: Because the extreme high values would artificially inflate the mean, you must use the median (location) and the interquartile range / IQR (spread).
3. Negatively Skewed (Skewed to the Left)
- The bulk of the data is bunched up on the right side of the histogram, with a long "tail" stretching out to the left.
- This happens when there are a few unusually low outliers pulling the data downwards.
- Appropriate Measures: Just like positive skew, you must use the median and the interquartile range.
4. Uniform Distribution
- The data is spread evenly across the entire range.
- A histogram of a uniform distribution looks relatively flat, with all the bars being roughly the exact same height.
Worked examples
Example 1
Example 1: Analysing Salaries (Positive Skew)
A researcher plots a histogram showing the annual salaries of 10,000 employees at a massive global tech company. They notice that the vast majority of employees earn between £20,000 and £40,000, but a small handful of senior executives earn over £500,000.
- (a) Describe the expected shape of this distribution.
- (b) State the most appropriate measure of location and measure of spread to summarise this data.
(a) Positively skewed (or skewed to the right) because the few massive salaries create a long tail to the right.
(b) The median and the interquartile range (IQR).
Example 2
Example 2: Exam Results (Negative Skew)
A driving instructor records the scores of 50 students on a basic road safety theory test out of 100. The test is considered very easy. 45 students score between 90 and 100, while 5 students score below 40.
- (a) Describe the shape of the distribution of these test scores.
- (b) An assistant calculates the mean score to summarise the data. Explain why this is not the most appropriate measure to use.
(a) Negatively skewed (or skewed to the left).
(b) The mean is not appropriate because the data is skewed. The very low scores of the 5 students will pull the mean down, making it an inaccurate representation of the typical student's high score. The median should be used instead.
Example 3
Example 3: Biological Data (Normal Distribution)
A marine biologist collects data on the lengths of 800 adult great white sharks. A histogram of the data shows a perfectly symmetrical bell shape.
State the most appropriate measure of location and measure of spread the biologist should use to summarise the shark lengths.
Because the data is symmetrically (normally) distributed, the biologist should use the mean and the standard deviation.
Example 4
Example 4: Dice Rolls (Uniform Distribution)
A student rolls a standard six-sided die 600 times and records the number of times each face lands face up. They plot a histogram of the results. Describe the likely shape of this distribution.
A uniform distribution.
(Since every number from 1 to 6 has an equal chance of landing, the bars for each number will be roughly the same height).