Statistics

The One Thing

Statistics uses numerical summaries and graphical representations to describe data, identify patterns and support conclusions. The choice of average, measure of spread or diagram must match the type, distribution and reliability of the data.

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 obtained from an existing source, such as a report, database or website.
• Discrete data: Countable numerical data that usually take separate values, such as the number of cars.
• Continuous data: Measured data that can take any value within a range, such as height, mass or time.
• Mean: The total of all data values divided by the number of values:

$$\text{mean}=\frac{\text{sum of values}}{\text{number of values}}.$$

• Median: The middle value when data are arranged in order. If there are two middle values, their mean is used.
• Mode: The value or category that occurs most frequently.
• Range: A simple measure of spread:

$$\text{range}=\text{maximum value}-\text{minimum value}.$$

• Quartile: A value that divides ordered data into four equal parts. The lower quartile is $Q_1$, and the upper quartile is $Q_3$.
• Interquartile range: The spread of the middle half of the data:

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

• Outlier: A value that is unusually high or low compared with the rest of the data.
• Frequency: The number of times a value or group of values occurs.
• Frequency table: A table listing values or class intervals with their frequencies.
• Grouped data: Data organised into class intervals rather than listed individually.
• Class width: The difference between the boundaries of a class interval.
• Frequency density: Frequency divided by class width:

$$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$$

• Histogram: A graph for continuous grouped data in which the area of each bar represents frequency. The vertical axis shows frequency density when class widths differ.
• Bar chart: A graph for discrete or categorical data with separate bars whose heights represent frequencies.
• Pie chart: A circular representation in which each sector angle is proportional to its frequency:

$$\text{sector angle}=\frac{\text{frequency}}{\text{total frequency}}\times 360^{\circ }.$$

• Stem-and-leaf diagram: A display that preserves individual values while showing their distribution. A key is required to explain the place values.
• Cumulative frequency: A running total of frequencies up to a particular value or class boundary.
• Cumulative frequency graph: A graph used to estimate medians, quartiles, percentiles and the number of values below a given value.
• Percentile: A value below which a stated percentage of the data lies.
• Scatter diagram: A graph in which paired values are plotted to investigate the relationship between two variables.
• Correlation: The degree to which two variables are related. It may be positive, negative or absent.
• Positive correlation: As one variable increases, the other generally increases.
• Negative correlation: As one variable increases, the other generally decreases.
• Line of best fit: A straight line representing the overall trend of a scatter diagram, with roughly balanced points on either side.
• Interpolation: Estimating a value within the range of observed data.
• Extrapolation: Estimating a value outside the observed data range; this is usually less reliable.
• Sample: A smaller group selected from a population to represent it.
• Bias: A systematic influence that causes collected data or a sample to be unrepresentative.

Worked Methods

Calculating averages and spread

• Mean from individual data

- Add all the data values. - Divide by the number of values: $$\text{mean}=\frac{\text{sum of values}}{\text{number of values}}.$$

• Mean from a frequency table

- Multiply each value by its frequency. - Add the products. - Divide by the total frequency: $$\text{mean}=\frac{\sum (\text{frequency}\times \text{value})}{\sum \text{frequency}}.$$

• Estimated mean from grouped data

- Find the midpoint of each class interval. - Multiply each class midpoint by its frequency. - Add the products. - Divide by the total frequency: $$\text{estimated mean}=\frac{\sum (\text{frequency}\times \text{class midpoint})}{\text{total frequency}}.$$

• Median

- Arrange the data in order. - If there is an odd number of values, identify the single middle value. - If there is an even number of values, calculate the mean of the two central values.

• Mode

- Identify the value or category with the greatest frequency.

• Range and interquartile range

- Calculate the range using: $$\text{range}=\text{maximum}-\text{minimum}.$$ - Order the data and identify $Q_1$ and $Q_3$ using the method specified in the question. - Calculate: $$\text{IQR}=Q_3-Q_1.$$ - The mean is usually more affected by outliers than the median. The median is generally more resistant to outliers.

Constructing and interpreting diagrams

• Frequency table

- List the individual values or class intervals. - Record the frequency for each value or interval. - For grouped data, calculate the class width where required.

• Bar chart

- Use discrete or categorical data. - Label both axes. - Draw separate bars whose heights represent frequencies.

• Histogram

- Use continuous grouped data. - Calculate frequency density: $$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$$ - Plot class intervals on the horizontal axis and frequency density on the vertical axis. - Draw touching bars, because continuous data have no gaps between class intervals. - Remember that the area of each bar represents frequency, especially when class widths are unequal.

• Pie chart

- Find the total frequency. - For each category, calculate: $$\text{sector angle}=\frac{\text{frequency}}{\text{total frequency}}\times 360^{\circ }.$$ - Draw and label the sectors. - If the sector angle is given, calculate frequency using: $$\text{frequency}=\frac{\text{sector angle}}{360^{\circ }}\times \text{total frequency}.$$

• Stem-and-leaf diagram

- Separate each value into a stem and a leaf according to place value. - Arrange the leaves in order. - Include a key explaining the place values. - Use the diagram to identify individual values and the distribution.

• Cumulative frequency graph

- Calculate the running total of frequencies. - Plot cumulative frequency against the upper class boundary. - Join the points with a smooth curve or suitable line. - Estimate: - the median at half the total frequency; - $Q_1$ at one-quarter of the total frequency; - $Q_3$ at three-quarters of the total frequency. - Estimate the interquartile range by reading: $$Q_3-Q_1.$$ - Use the graph to estimate percentiles and the number of values below a given value. These estimates are not exact when the data have been grouped.

Scatter diagrams and correlation

• Plot each pair of values accurately.
• Label both axes and use an appropriate scale.
• Examine the overall pattern:

- positive correlation means both variables generally increase together; - negative correlation means one variable generally decreases as the other increases; - absent correlation means there is no clear relationship.

• Draw a line of best fit to represent the general trend.
• Keep roughly equal numbers of points on either side of the line where possible.
• Do not force the line through the origin unless there is a mathematical reason.
• Use interpolation to estimate within the observed data range.
• Use extrapolation to estimate outside the observed range, recognising that it is usually less reliable.
• Treat correlation as association, not proof of causation: a third factor may influence both variables.

Sampling and interpretation

• Select a sample from the population.
• Consider whether the sample is fairly selected and representative.
• Check for bias, which may make the sample or data unrepresentative.
• A larger, fairly selected sample is generally more reliable than a small or biased sample, although sampling error may still occur.
• Interpret the centre, spread, shape of the distribution, possible outliers, sample size, source of data and graph scale.
• Check for shortened or uneven axes, since these can exaggerate differences.

Where It Goes Wrong

• Calculating the mean without ordering the data first when finding the median or quartiles; the median requires ordered data.
• Using maximum minus minimum for the interquartile range; the range is maximum minus minimum, whereas $\text{IQR}=Q_3-Q_1$.
• Treating grouped-data means as exact; a grouped-data mean is an estimated mean based on class midpoints.
• Drawing histogram bars with equal visual importance when class widths differ; frequency density must be used, and bar area represents frequency.
• Separating histogram bars or joining bar-chart bars; histogram bars touch for continuous data, while bar-chart bars are separated for distinct categories or values.
• Assuming correlation proves causation, or forcing a line of best fit through the origin without a mathematical reason.
• Reading cumulative-frequency values as exact rather than estimated, or ignoring the upper class boundary when plotting.
• Accepting a graph or conclusion without checking its scale, labels, sample size, source of data, possible bias and outliers.

What Gets Asked

• Calculate the mean, median, mode, range, quartiles and interquartile range from raw or frequency-table data.
• Explain which average is most suitable for balanced, skewed or outlier-containing data.
• Find an estimated mean from grouped data using class midpoints.
• Construct or interpret frequency tables, bar charts, histograms, pie charts and stem-and-leaf diagrams.
• Calculate pie-chart sector angles or frequencies.
• Calculate frequency density and use it to construct or interpret histograms with unequal class widths.
• Construct cumulative-frequency graphs and estimate the median, quartiles, percentiles, interquartile range or number of values below a given value.
• Plot and interpret scatter diagrams, identify positive, negative or absent correlation, and draw a line of best fit.
• Use interpolation and extrapolation to make estimates.
• Explain why correlation does not establish causation.
• Assess the reliability of a sample, identify bias and discuss sampling error.
• Critique misleading graphs by examining their scales, labels and axes.
• Compare data sets using both an average and a measure of spread, while considering the context, distribution shape and possible outliers.