Statistics

The One Thing

Statistics uses appropriate data collection, displays and summary measures to describe the centre, spread and shape of data. Reliable conclusions require consideration of variation, outliers, bias, sample representation, context and the limitations of the evidence.

Definitions and Results

• Population: The complete group of people, objects or measurements being studied.
• Sample: A smaller group selected from a population to provide information about the whole population.
• Variable: A characteristic that can take different values, such as height, travel time or favourite sport.
• Categorical data: Data recorded as categories or labels, such as eye colour or transport type.
• Numerical data: Data recorded as numbers that represent counts or measurements.
• Discrete data: Numerical data counted in separate whole-number values, such as the number of pets.
• Continuous data: Numerical data measured on a scale and able to take any value within an interval, such as mass or temperature.
• Mean: The total of all data values divided by the number of values; it represents a balance point.

$$\text{mean}=\frac{x_1+x_2+\cdots +x_n}{n}$$ The mean uses every value and is sensitive to outliers.

• Median: The middle value when data is arranged in order. If there are two middle values, their average is used. The median is usually more resistant to outliers than the mean.
• Mode: The value or category that occurs most often.
• Range: The difference between the largest and smallest values.

$$\text{range}=\text{maximum value}-\text{minimum value}$$ It uses only the two extreme values and can be strongly affected by an unusual minimum or maximum.

• Quartile: A value that divides ordered data into parts. The lower quartile, $Q_1$, has about 25% of values below it, and the upper quartile, $Q_3$, has about 75% below it.
• Interquartile range (IQR): The spread of the middle 50% of the data.

$$\text{IQR}=Q_3-Q_1$$ It is generally less affected by extreme values than the range.

• Five-number summary: The minimum, lower quartile, median, upper quartile and maximum.
• Box plot: A graph based on the five-number summary that displays the centre, spread and possible outliers.
• Histogram: A graph for numerical data grouped into intervals, with adjoining bars whose areas or heights represent frequencies. For unequal class widths, frequency density should be considered so that bar area represents frequency fairly.
• Outlier: A value unusually far from the rest of the data. It may affect summaries, especially the mean and range, and should be investigated rather than automatically removed.
• Distribution: The overall pattern of data, including its centre, spread, shape, clusters and gaps.
• Skewness: A lack of symmetry in a distribution. A right-skewed distribution has a longer tail toward larger values, whereas a left-skewed distribution has a longer tail toward smaller values.
• Association: A relationship or pattern between two variables. Association does not by itself prove that one variable causes the other.
• Bias: A systematic problem in the way data is collected, selected or reported that can make conclusions unfair or inaccurate.
• Relative frequency: The proportion or percentage of observations in a category or interval.

$$\text{relative frequency}=\frac{\text{frequency}}{\text{total frequency}}$$

• Frequency table: A table recording how often each category or value occurs.
• Relative-frequency table: A table recording the proportion or percentage of observations in each category or interval.
• Bar chart: A display suitable for categorical or discrete data. Bars usually have gaps between them.
• Statistical investigation: A process involving a question, data collection, appropriate displays, numerical summaries, interpretation and a conclusion with limitations.
• Central comparison principle: No single summary measure describes every data set well. The choice depends on the data type, distribution, outliers and purpose.
• Variation principle: A larger spread indicates greater variation. Two groups may have similar centres but very different variability.
• Comparison principle: Comparisons should use the same type of summary and consider both centre and spread.
• Sampling principle: A representative sample should reflect the population. Voluntary-response, convenience and poorly selected samples may produce biased results. A larger sample does not automatically remove bias.
• Interpretation principle: Percentages and graphs must be interpreted using their scales, sample sizes and context rather than visual appearance alone.
• Causation principle: Association between variables should not be treated as proof of causation because other variables may influence the results.

Worked Methods

Calculating the mean

• Add all data values.
• Count the number of values.
• Divide the total by the number of values.
• Report the result in the appropriate units.

Use: $$\text{mean}=\frac{x_1+x_2+\cdots +x_n}{n}$$

The mean incorporates every value, but an outlier can substantially alter it.

Finding the median

• Arrange the data values in ascending order.
• If there is one middle value, use it as the median.
• If there are two middle values, calculate their average.
• Interpret the median as dividing the ordered data into two equal parts.

Finding the mode

• Record the frequency of each value or category.
• Identify the value or category with the greatest frequency.
• Report it as the mode.

A data set may have more than one mode or no mode.

Calculating the range

• Identify the maximum value.
• Identify the minimum value.
• Subtract the minimum from the maximum.

Use: $$\text{range}=\text{maximum value}-\text{minimum value}$$

Finding quartiles and the interquartile range

• Arrange the data in order.
• Find the median.
• Divide the data into the lower and upper parts.
• Find the lower quartile, $Q_1$, from the lower part.
• Find the upper quartile, $Q_3$, from the upper part.
• Calculate:

$$\text{IQR}=Q_3-Q_1$$

The quartiles divide ordered data into four parts, and the IQR describes the middle 50%.

Constructing a five-number summary and box plot

• Find the minimum.
• Find $Q_1$.
• Find the median.
• Find $Q_3$.
• Find the maximum.
• Record the five-number summary:

$$\text{minimum},Q_1,\text{median},Q_3,\text{maximum}$$

• Use a suitable scale.
• Draw the box from $Q_1$ to $Q_3$.
• Mark the median inside the box.
• Extend the whiskers towards the minimum and maximum, while identifying possible outliers where appropriate.

Use box plots to compare medians, IQRs, overall ranges and possible outliers between groups.

Choosing between the mean and median

• Examine the distribution for symmetry, skewness and outliers.
• Use the mean when all values should contribute and the data has no strong outliers.
• Use the median when the data is skewed or contains influential outliers.
• Compare the mean and median where useful:

- For symmetric data without strong outliers, they are often similar. - In skewed data, the median may better represent a typical value.

• Report a measure of spread as well as the centre.

Constructing and interpreting frequency displays

• Identify whether the data is categorical, discrete or continuous.
• Use a frequency table to record how often each category or value occurs.
• Calculate relative frequency when proportions or percentages are required:

$$\text{relative frequency}=\frac{\text{frequency}}{\text{total frequency}}$$

• Use a bar chart for categorical or discrete data, leaving gaps between bars.
• Use a histogram for grouped continuous data, with adjoining bars.
• If class intervals have unequal widths, use frequency density so that bar area represents frequency fairly.
• Interpret the display using its scale, sample size and context.

Comparing groups

• Use the same type of summary for each group.
• Compare centres using the mean or median.
• Compare variation using the range or IQR.
• Examine box plots for medians, IQRs, overall ranges and possible outliers.
• Consider the shape, clusters, gaps and skewness of each distribution.
• State whether one group is more variable, even if the groups have similar centres.
• Interpret the comparison in context and acknowledge limitations.

Conducting a statistical investigation

• Formulate a question.
• Identify the population, sample and variables.
• Collect data using a method that aims to produce a representative sample.
• Check for possible voluntary-response, convenience or selection bias.
• Classify the data as categorical, numerical, discrete or continuous.
• Select appropriate displays, such as frequency tables, relative-frequency tables, bar charts, histograms or box plots.
• Calculate suitable summaries, including the mean, median, mode, range, quartiles and IQR.
• Examine centre, spread, shape, clusters, gaps, skewness and outliers.
• Interpret the results using units, scale, sample size and context.
• Draw a conclusion supported by evidence and state the investigation’s limitations.
• Do not infer causation solely from an association.

Where It Goes Wrong

• Treating the mean, median, mode, range and IQR as interchangeable, rather than selecting a measure according to the data type, distribution, outliers and purpose.
• Forgetting to arrange data before finding the median or quartiles, or forgetting that two middle values require an average.
• Reporting the range without recognising that it uses only the minimum and maximum and may be distorted by an unusual value.
• Using a bar chart for grouped continuous data or a histogram for categorical or discrete data; histograms require adjoining bars.
• Ignoring unequal class widths in a histogram instead of considering frequency density so that bar area represents frequency fairly.
• Drawing conclusions from a larger sample without checking whether the selection method produced a representative sample, or interpreting association as proof of causation.

What Gets Asked

• Define population, sample, variable, categorical data, numerical data, discrete data and continuous data.
• Calculate and interpret the mean, median, mode and range.
• Find quartiles and calculate the interquartile range using $Q_3-Q_1$.
• Construct or interpret a five-number summary and box plot.
• Compare two groups using centre, spread, IQR, range, possible outliers and distributional features.
• Select an appropriate display: frequency table, relative-frequency table, bar chart, histogram or box plot.
• Interpret histograms with equal or unequal class intervals, including frequency density.
• Identify skewness, clusters, gaps, variation and outliers.
• Explain why the median or IQR may be preferable to the mean or range.
• Evaluate whether a sample is representative and identify voluntary-response, convenience or other selection bias.
• Interpret percentages and graphs using their scales, sample sizes and context.
• Distinguish association from causation.
• Plan or evaluate a statistical investigation, including its question, data collection, displays, summaries, conclusion and limitations.