Measures of central tendency and spread, histograms, box plots and cumulative frequency, identifying outliers, comparing distributions, and correlation and the regression line.

What this dot point is asking

OCR wants you to calculate and interpret measures of central tendency (mean, median, mode) and spread (range, interquartile range, variance, standard deviation), draw and read histograms (with frequency density), box plots and cumulative frequency graphs, identify outliers by a stated rule, compare two distributions, and interpret correlation and the least-squares regression line, including the dangers of extrapolation.

The answer

Averages and spread

The mean uses every value but is sensitive to outliers; the median is the middle value and is resistant to outliers; the mode is the most common value. The standard deviation measures typical distance from the mean, while the interquartile range measures the spread of the middle half and ignores extremes.

Histograms and frequency density

A histogram shows grouped continuous data with area proportional to frequency, so the vertical axis is frequency density, not frequency.

Outliers

An outlier is a value far from the rest. Two common rules are: more than $1.5 \times \text{IQR}$ beyond a quartile, or more than two standard deviations from the mean. The question states which rule to use.

Box plots and skewness

A box plot shows the minimum, the three quartiles and the maximum. Comparing the median's position within the box describes skewness: a median nearer $Q_1$ indicates positive skew, nearer $Q_3$ indicates negative skew.

Examples in context

Comparing two distributions

To compare data sets, always compare a measure of location and a measure of spread, in context. For example "the median mark of class A (62) is higher than class B (55), and class A's smaller interquartile range (10 versus 18) shows its marks were more consistent."

Correlation and regression

Correlation measures how closely two variables follow a linear relationship; the product moment correlation coefficient $r$ runs from $-1$ to $1$. The regression line of $y$ on $x$ is the best-fit line for predicting $y$ from $x$. Use it only within the range of the data: predicting outside that range (extrapolation) is unreliable, and correlation alone does not prove that one variable causes the other.

Try this

Q1. A class width is $5$ and the frequency is $30$. Find the frequency density. [1 mark]

• Cue. $\dfrac{30}{5} = 6$.

Q2. Data has mean $50$ and standard deviation $4$. Using the two-standard-deviation rule, find the upper outlier boundary. [2 marks]

• Cue. $50 + 2(4) = 58$.