Data interpretation

What is single-variable (univariate) data?

Data consisting of observations of one variable, e.g. heights of students.

What is bivariate data?

Data consisting of paired observations of two variables, e.g. (height, weight) for each student.

What does a vertical line chart show?

Frequencies of discrete data, with the height of each vertical line representing the frequency.

What does a dot plot show?

Each individual data value as a dot above its value on a number line; useful for small data sets.

What does a stem-and-leaf diagram show?

Each data value split into a stem and a leaf, so the raw data is preserved while showing the distribution.

What is a back-to-back stem-and-leaf diagram used for?

Comparing two single-variable data sets that share the same stems.

What does a box-and-whisker plot show?

The minimum, lower quartile, median, upper quartile and maximum (the five-number summary), and any outliers.

What does the box in a box plot represent?

The interquartile range — from the lower quartile (Q1) to the upper quartile (Q3), with the median marked inside.

What does a cumulative frequency diagram show?

The running total of frequencies up to each value, plotted at upper class boundaries; useful for finding the median, quartiles and percentiles.

In a histogram, what does the area of a bar represent?

Frequency (or relative frequency).

What is plotted on the vertical axis of a histogram with unequal class widths?

Frequency density, where frequency density = frequency ÷ class width.

How do you find the frequency of a class from a histogram?

Multiply the frequency density of that class by the class width.

How are histograms linked to continuous probability distributions?

As class widths shrink, the histogram of relative frequencies approaches the probability density function of the distribution.

What is a scatter diagram?

A diagram showing bivariate data as points (x, y), used to look for relationships between two variables.

What is a regression line?

A straight line fitted through the points of a scatter diagram, modelling the relationship between the two variables (calculation of its equation is not required at A-level).

What does positive correlation mean?

As one variable increases, the other tends to increase.

What does negative correlation mean?

As one variable increases, the other tends to decrease.

What does no correlation mean?

No clear linear relationship between the two variables.

Why does correlation not imply causation?

A correlation between two variables can arise by chance, from reverse causation, or from a hidden third variable; it does not by itself show that one variable causes the other.

What is the mean?

The sum of the data values divided by the number of values. The mean of x̄ = (Σx) / n.

What is the median?

The middle value when the data are ordered (the average of the two middle values if n is even).

What is the mode?

The value (or values) that occur most frequently in the data set.

What is a percentile?

The value below which a given percentage of the data falls; e.g. the 30th percentile is the value below which 30% of the data lie.

What is the lower quartile (Q1)?

The 25th percentile — the value a quarter of the way through the ordered data.

What is the upper quartile (Q3)?

The 75th percentile — the value three-quarters of the way through the ordered data.

What is the interquartile range (IQR)?

IQR = Q3 − Q1; a measure of spread that ignores the extremes.

What is the variance?

The mean of the squared deviations from the mean: σ² = Σ(x − x̄)² / n = Σx²/n − x̄².

What is the standard deviation?

The square root of the variance — the root-mean-square deviation from the mean.

Formula for standard deviation from a list of data

σ = √(Σ(x − x̄)² / n) = √(Σx²/n − x̄²).

Formula for standard deviation from a frequency distribution

σ = √(Σf(x − x̄)² / Σf) = √(Σfx²/Σf − x̄²).

Why are the mean and standard deviation from a grouped frequency table only estimates?

Because exact values within each class are unknown; the class midpoint is used as an approximation.

How can you use mean and standard deviation to compare distributions?

Compare means to compare averages and compare standard deviations to compare spread/consistency, in the context of the data.

Outlier rule 1 (IQR rule)

A data point is an outlier if it is more than 1.5 × IQR below Q1 or above Q3.

Outlier rule 2 (standard deviation rule)

A data point is an outlier if it is more than 2 standard deviations away from the mean.

What is data cleaning?

Preparing a data set for analysis by handling missing values, correcting errors and deciding what to do with outliers.

When might it be appropriate to remove an outlier?

When it is shown to be a measurement or recording error, or when it clearly does not belong to the population of interest.

When should an outlier be kept?

When it is a genuine extreme observation that gives meaningful information about the population.