Revise Descriptive statistics for IB Mathematics: Applications and Interpretation (4.2)

A measure of central tendency is a single value that stands in for a whole data set, a typical value. The three you need are the mean, the median and the mode. The mean uses every value, so it carries the most information but is the most easily distorted by an extreme value. The median, the middle value once the data is ordered, ignores how extreme the ends are and so resists outliers. The mode is simply the most common value, and is the only one of the three that works for non-numerical data such as favourite colour.

Mean of a data set: $\bar{x} = \dfrac{\sum_{i=1}^{k} f_i x_i}{n}$, where $n = \sum_{i=1}^{k} f_i$. \n For raw data every $f_i = 1$; for grouped data $x_i$ is the mid-interval value of each class.

The median sits at position $\frac{n+1}{2}$ in the ordered list. If $n$ is odd this is a single value; if $n$ is even it falls between two values, and you average them. A frequent slip is to read the median off unordered data, so always order first, or let the GDC do it.

Knowing the centre is only half the story. Two classes can share a mean score of 60% while one is full of steady 55-to-65 marks and the other swings between 30 and 90. Dispersion measures how spread out the data is. The three you need are the range, the interquartile range and the standard deviation.

Quartiles cut ordered data into four equal parts. The lower quartile $Q_1$ has 25% of the data below it, the median $Q_2$ has 50%, and the upper quartile $Q_3$ has 75%. Percentiles generalise this to hundredths: the $p$th percentile is the value below which $p\%$ of the data lies, so $Q_1$, $Q_2$ and $Q_3$ are just the 25th, 50th and 75th percentiles.

For grouped data, percentiles are usually read off a cumulative frequency graph. Cumulative frequency is a running total: for each class you add its frequency to all those before it, then plot the cumulative frequency against the upper boundary of the class. The result is the familiar S-shaped curve. To estimate the median of 200 data points, go up to a cumulative frequency of 100 and read across; for the 60th percentile, go up to 120 and read across; and $Q_1$ and $Q_3$ come from cumulative frequencies of 50 and 150. Reading across then down converts a height on the curve into a data value.

A box-and-whisker plot draws the five-number summary, minimum, $Q_1$, median, $Q_3$, maximum, on a common scale. The box stretches from $Q_1$ to $Q_3$ (its width is the IQR) with the median marked inside; the whiskers reach out to the most extreme values that are not outliers. Any genuine outliers are plotted as separate points beyond the whiskers. The picture reveals skew at a glance: a median sitting closer to $Q_1$ than to $Q_3$, or a long right whisker, signals data stretched towards the high end.

An outlier is a value far enough from the bulk of the data to deserve suspicion. The IB uses a precise test rather than a hunch. Build two 'fences': the lower fence at $Q_1 - 1.5\,\text{IQR}$ and the upper fence at $Q_3 + 1.5\,\text{IQR}$. Any value beyond a fence is an outlier. The rule is deliberately mechanical, so you must compute the fence and compare, never simply declare the largest value an outlier because it looks big.

Lower fence: $Q_1 - 1.5 \times \text{IQR}$ \n Upper fence: $Q_3 + 1.5 \times \text{IQR}$ \n A value is an outlier if it is below the lower fence or above the upper fence.

Data often arrives grouped into class intervals, where you know how many values fall in each class but not the exact values. To summarise it you make one assumption: treat every value in a class as sitting at the class's mid-interval value (the midpoint of its endpoints). With that, the GDC estimates the mean and standard deviation. Because you no longer have the individual values, the answers are estimates, and the question will usually signal this with the word 'estimate'. There is no single mode either, only a modal class: the interval with the highest frequency.

Examiners love to ask what happens to the statistics when the data is transformed, because you can answer instantly if you understand centre versus spread. Adding a constant slides the whole data set along the number line: every location statistic moves with it, but nothing is stretched, so the spread is untouched. Multiplying by a constant stretches the data about zero: locations scale by the factor, and so does the spread.

For raw data, enter the values in one list and run one-variable statistics; the screen gives you $\bar{x}$, the median 'Med', $n$, the quartiles $Q_1$ and $Q_3$, the minimum, the maximum, and both standard deviations $\sigma_x$ and $s_x$. For frequency or grouped data, put the values (or mid-interval values) in one list, the frequencies in a second, and tell the calculator to use the second list as frequencies. Two habits save marks: take the population standard deviation $\sigma_x$ unless told 'sample', and read the quartiles for the IQR rather than trying to locate them by hand.

In Paper 2 the marks are analytic: each is tied to a specific line of working, a method mark (M) or an accuracy mark (A), and an accuracy mark depends on the method mark it follows. Follow-through (FT) means a value that is wrong earlier does not have to cost you the marks that depend on it, provided the later step is done correctly on your figure. The engine also ignores subsequent working (ISW) once a correct answer appears, and accepts any equivalent form and any correctly-rounded value. But that protection only exists if the method is on the page. Study how each mark below is earned by a specific line.

Descriptive statistics is the vocabulary the rest of the statistics course speaks. The mean and standard deviation you read here reappear as the parameters of the normal distribution, where the standard deviation sets the width of the bell; quartiles and percentiles return as probabilities you read off that curve. The idea of a robust versus a sensitive measure shapes how you interpret correlation and regression, and the habit of showing method, subtracting quartiles and testing before concluding is exactly the discipline every Paper 2 statistics question rewards. Master the two headlines, a centre and a spread chosen to suit the data, and the statistics that follows becomes variations on a summary you already know how to write.