Describing Distributions

Types of Distributions

A distribution shows how data values are spread out or arranged. Understanding the shape of a distribution helps interpret patterns in geographical data.

• Symmetrical distribution: Data is evenly spread around the centre. The left and right sides are mirror images. The mean, median, and mode are all close or the same. Example: Heights of a group of people.
• Skewed distribution: Data is not symmetrical and leans towards one side.
  • Positive skew (right skew): Most data is on the left with a long tail to the right. The mean is greater than the median. Example: Income distribution where a few people earn very high wages.
  • Negative skew (left skew): Most data is on the right with a long tail to the left. The mean is less than the median. Example: Age at retirement where most retire around 65 but some retire very early.
• Uniform distribution: All data values occur with roughly equal frequency. The graph looks flat. Example: Rolling a fair six-sided die many times.
• Bimodal distribution: Two distinct peaks or modes appear, showing two common values or groups. Example: Temperatures in a city with two seasons, summer and winter.

Describing Distribution Shape

When describing a distribution, focus on these key features:

• Peaks and modes: The highest points on the graph show where data values occur most often. A mode is the value with the highest frequency. A distribution can have one mode (unimodal), two modes (bimodal), or more.
• Spread and range: Spread shows how far data values are from each other. The range is the difference between the highest and lowest values. A wide range means data is spread out; a narrow range means data is clustered.
• Outliers: Values that are much higher or lower than the rest of the data. Outliers can affect averages and may indicate unusual events or errors.
• Clusters: Groups of data points close together. Clusters show concentrations of values in certain areas.

For example, a histogram showing rainfall in a UK region might have one peak (mode) around 50 $\mathrm{mm}$, a range from 10 $\mathrm{mm}$ to 100 $\mathrm{mm}$, and an outlier at 150 $\mathrm{mm}$ due to a rare storm.

Measures of Central Tendency

Central tendency measures summarise a set of data by identifying a central or typical value. The three main measures are:

• Mean: The average value. Add all data points and divide by the number of points. Useful for data without extreme outliers.
• Median: The middle value when data is ordered from smallest to largest. Useful when data is skewed or has outliers.
• Mode: The most frequently occurring value. Useful for categorical data or to identify common values.

When to use each:

• Use the mean for symmetrical data without outliers.
• Use the median for skewed data or when outliers are present.
• Use the mode to find the most common category or value.

For instance, if you have the following data on daily temperatures in a city ($^{\circ}\mathrm{C}$): 12, 14, 14, 15, 16, 18, 50 (an outlier), the mean would be pulled up by 50, so the median (15) better represents the typical temperature.

Interpreting Graphs and Data

Different graphs help visualise data distributions. Key types include:

• Histograms: Show frequency of data within intervals (bins). Bars touch each other, indicating continuous data. Useful for showing distribution shape, spread, and outliers.
• Bar charts: Show frequency of categories with separate bars. Bars do not touch. Useful for comparing discrete categories like land use types.
• Scatter graphs: Plot pairs of numerical data points to show relationships or correlations. Points may form patterns or clusters.
• Line graphs: Connect data points with lines, often showing trends over time or distance.

When interpreting these graphs, describe the shape of the distribution, central tendency, spread, and any unusual features like outliers or clusters.

For example, a scatter graph showing rainfall and crop yield might show a positive correlation, with points clustered along a rising line.

Learning example: If a histogram of daily rainfall ($\mathrm{mm}$) has bars highest around 5–10 $\mathrm{mm}$ and a long tail stretching to 50 $\mathrm{mm}$, the distribution is positively skewed with a peak (mode) at 5–10 $\mathrm{mm}$ and some outliers at higher rainfall.