Statistics

The One Thing

Statistics uses appropriate methods to collect, organise, represent and interpret data. Reliable conclusions depend on representative sampling, suitable diagrams and measures of centre and spread, while allowing for bias, outliers, scale and the distinction between correlation and causation.

Definitions and Results

• Data: Information collected for analysis. It may be numerical or categorical.
• Primary data: Data collected directly for a particular investigation, such as through a survey or experiment.
• Secondary data: Data collected by someone else and used for a new purpose.
• Qualitative data: Non-numerical data describing categories or qualities, such as colour or type of transport.
• Quantitative data: Numerical data that can be measured or counted.
• Discrete data: Countable numerical data, usually taking separate values, such as the number of pets.
• Continuous data: Measured numerical data that can take any value within a range, such as height or time.
• Population: The complete group being studied.
• Sample: A smaller group selected from the population.
• Census: A data collection method that includes every member of the population.
• Random sample: A sample selected so that each member of the population has an equal or known chance of being chosen.
• Systematic sample: A sample selected using a fixed pattern, such as choosing every tenth person.
• Stratified sample: A sample that reflects the proportions of important groups within the population.
• Bias: A systematic tendency for data collection or presentation to favour a particular result.
• Frequency: The number of times a value or category occurs.
• Frequency table: A table showing values or categories alongside their frequencies.
• Two-way table: A table used to display the relationship between two categorical variables.
• Bar chart: A diagram using separated bars to represent frequencies for discrete or categorical data.
• Pie chart: A circular diagram in which each sector represents a category’s proportion of the total.
• Histogram: A graph for grouped continuous data in which bars touch and area represents frequency.
• Frequency density: The frequency per unit width of a class interval.
• Frequency polygon: A graph formed by joining points representing class midpoints and frequencies or frequency densities.
• Scatter diagram: A graph showing paired numerical data to investigate a possible relationship.
• Correlation: A relationship between two variables. It may be positive, negative or absent, and can be strong or weak.
• Line of best fit: A straight line drawn to represent the overall trend in a scatter diagram.
• Mean: The total of all values divided by the number of values.
• Median: The middle value when data is arranged in order; if there are two middle values, their mean is used.
• Mode: The value or category occurring most often.
• Range: The difference between the largest and smallest values.
• Quartile: A value that divides ordered data into four approximately equal parts.
• Interquartile range: The difference between the upper quartile and lower quartile, measuring the spread of the middle half of the data.
• Outlier: A value noticeably distant from the rest of the data.
• Cumulative frequency: A running total of frequencies up to each value or class boundary.
• Box plot: A diagram showing the minimum, lower quartile, median, upper quartile and maximum.
• Estimated mean: An approximate mean calculated from grouped data using class midpoints.

Important results:

• Mean = sum of values ÷ number of values.
• For a frequency table:

mean = sum of (value × frequency) ÷ sum of frequencies.

• Estimated mean for grouped data:

estimated mean = sum of (class midpoint × frequency) ÷ total frequency.

• Range = maximum value − minimum value.
• Interquartile range = upper quartile − lower quartile.
• For a pie chart:

sector angle = frequency ÷ total frequency × 360°.

• For a histogram:

frequency density = frequency ÷ class width.

• Histogram bar area represents frequency; therefore, unequal class widths require frequency density.
• Class midpoint = (lower class boundary + upper class boundary) ÷ 2.
• To find a median or quartile from a cumulative frequency graph, use the relevant position in the ordered data and read across to the graph.
• A sample should be representative of the population. A larger, well-chosen sample generally gives more reliable results.
• The mean uses every value and is useful for balanced data, but it is affected by outliers.
• The median is less affected by outliers and is often better for skewed data.
• The mode identifies the most common category or value but may not exist or may not be unique.
• Averages describe the centre of data, whereas range and interquartile range describe spread.
• Correlation does not prove that one variable causes another; a third variable may explain the relationship.
• Graphs should have suitable scales, labelled axes, clear units and an appropriate title.
• A misleading graph may use a non-zero vertical axis, unequal intervals or an inappropriate diagram type.
• When comparing data sets, consider both the average and the spread, and interpret the results in context.

Worked Methods

1. Selecting and assessing data

• Identify whether the data is primary or secondary.
• Classify it as qualitative or quantitative.
• If it is quantitative, decide whether it is discrete or continuous.
• Identify the population being studied.
• Decide whether a census or sample is appropriate.
• If using a sample, select a random, systematic or stratified sample.
• Check whether the sample is representative and whether the method or question introduces bias.
• Consider sample size: a larger, well-chosen sample generally produces more reliable results.

Examples of sampling methods include choosing every tenth person for a systematic sample and ensuring that important groups occur in the same proportions as in the population for a stratified sample.

2. Constructing frequency tables and two-way tables

• List the values or categories.
• Count the number of occurrences of each value or category.
• Record these counts as frequencies in a frequency table.
• For two categorical variables, place one variable in the rows and the other in the columns.
• Enter the frequencies for each combination of categories.
• Use row and column totals where required to examine the relationship between the variables.

3. Constructing and interpreting a bar chart

• Use a bar chart for discrete or categorical data.
• Place categories or discrete values on the horizontal axis.
• Place frequency on the vertical axis.
• Choose a suitable, even scale.
• Draw separated bars with heights equal to the frequencies.
• Label both axes, include units where relevant and give the diagram an appropriate title.
• Interpret the most and least frequent categories and compare their frequencies.

The bars are separated because the data is discrete or categorical.

4. Constructing a pie chart

• Find the total frequency.
• Calculate each sector angle using

sector angle = frequency ÷ total frequency × 360°.

• Draw a circle and measure each calculated angle.
• Label the sectors clearly and include a key if necessary.
• Check that the sector angles total 360°.
• Interpret each sector as a category’s proportion of the total.

5. Constructing a histogram

• Use a histogram for grouped continuous data.
• Place class intervals on the horizontal axis.
• Calculate the class width for each interval.
• Calculate frequency density using

frequency density = frequency ÷ class width.

• Draw touching bars with heights equal to the frequency densities.
• Use suitable class boundaries and scales, and label the axes and units.
• Interpret frequency through the area of each bar, not simply its height.

If class widths are unequal, frequency density must be used because histogram bar area represents frequency.

6. Constructing a frequency polygon

• Identify each class interval.
• Calculate each class midpoint using

class midpoint = (lower class boundary + upper class boundary) ÷ 2.

• Plot each midpoint against its frequency or frequency density, as appropriate.
• Join the plotted points with straight line segments.
• Interpret the resulting shape as a representation of the distribution.

7. Constructing and interpreting a scatter diagram

• Identify the two numerical variables.
• Place one variable on the horizontal axis and the other on the vertical axis.
• Plot each pair of values as a point.
• Use suitable scales, labelled axes, clear units and an appropriate title.
• Describe the direction of any correlation as positive, negative or absent.
• Describe its strength as strong or weak.
• Draw a line of best fit if an overall trend is required.
• Interpret the relationship in context.
• Do not conclude that one variable causes the other: a third variable may explain the relationship.

8. Calculating the mean from raw data

• Add all the values.
• Count the number of values.
• Divide the total by the number of values:

mean = sum of values ÷ number of values.

• State the answer in the context and units of the data where appropriate.

9. Calculating the mean from a frequency table

• Multiply each value by its frequency.
• Add the resulting products.
• Add the frequencies.
• Divide the sum of the products by the sum of the frequencies:

mean = sum of (value × frequency) ÷ sum of frequencies.

10. Calculating an estimated mean from grouped data

• Find the class midpoint for each interval:

class midpoint = (lower class boundary + upper class boundary) ÷ 2.

• Multiply each class midpoint by its frequency.
• Add these products.
• Add the frequencies to obtain the total frequency.
• Divide by the total frequency:

estimated mean = sum of (class midpoint × frequency) ÷ total frequency.

• Recognise that the result is an approximation because individual values within each class are not known.

11. Finding the median and quartiles

• Arrange raw data in order.
• Find the middle value for the median.
• If there are two middle values, calculate their mean.
• Identify the lower and upper quartiles as values dividing the ordered data into four approximately equal parts.
• For grouped or cumulative data, use the relevant position in the ordered data.
• On a cumulative frequency graph, read across from the required position to the graph and then down to the data value.

12. Calculating range and interquartile range

• Identify the maximum and minimum values.
• Calculate

range = maximum value − minimum value.

• Identify the upper and lower quartiles.
• Calculate

interquartile range = upper quartile − lower quartile.

• Use range to describe the total spread and interquartile range to describe the spread of the middle half.

13. Constructing and interpreting a box plot

• Determine the minimum value.
• Determine the lower quartile.
• Determine the median.
• Determine the upper quartile.
• Determine the maximum value.
• Mark these five values on a suitable scale.
• Draw the box from the lower quartile to the upper quartile, with a line at the median.
• Extend whiskers to the minimum and maximum.
• Compare the medians and spreads when comparing data sets.

Where It Goes Wrong

• Treating a sample as reliable without checking whether it is representative of the population, whether it is sufficiently large or whether the collection method introduces bias.
• Using a bar chart for continuous grouped data, or using a histogram without touching bars and without calculating frequency density for unequal class widths.
• Forgetting that histogram area, rather than height alone, represents frequency.
• Calculating a mean from a frequency table without multiplying each value by its frequency, or calculating an estimated mean without using class midpoints.
• Finding a median or quartile without first considering the ordered positions, or failing to read the relevant position correctly from a cumulative frequency graph.
• Interpreting a correlation as proof of causation, or accepting a graph without checking its scale, intervals, labels, units and suitability.

What Gets Asked

This material supports questions requiring students to:

• Classify data as primary or secondary, qualitative or quantitative, and discrete or continuous.
• Identify the population, sample and appropriate sampling method, including census, random, systematic and stratified sampling.
• Explain or identify possible bias and assess whether a sample is representative.
• Complete frequency tables and two-way tables.
• Draw, interpret or critique bar charts, pie charts, histograms, frequency polygons, scatter diagrams, cumulative frequency graphs and box plots.
• Calculate pie-chart sector angles, histogram frequency densities and class midpoints.
• Calculate the mean from raw data or a frequency table and the estimated mean from grouped data.
• Find the median, mode, range, quartiles and interquartile range.
• Use cumulative frequency graphs to estimate medians and quartiles.
• Compare data sets using averages and measures of spread.
• Describe correlation as positive, negative or absent and strong or weak, and draw or use a line of best fit.
• Explain why correlation does not establish causation.
• Interpret statistical results in context while considering outliers, sample size, scale, spread, bias and limitations.