Averages and spread

Median, quartiles, mean and standard deviation

In 6 of 20 recent exams
Paper 1 or 2
1 to 3 marks

How it is set in the exam

Paper 1 (no calculator) and Paper 2 (calculator), 1–3 marks a part: the opening parts of a statistics question, often several 1-mark "write down" parts in a row. Common (6 of 20 exam versions, 2 of them on Paper 1 (no calculator); 24 logged parts). On Paper 1 (no calculator) the list is short and the positions are counted by hand; on Paper 2 (calculator) everything comes from one-variable statistics, and the variance is the step with a method mark.

Spot it in the exam

Exam phrases to spot (9)

“find the median”
“find the interquartile range”
“write down the mode”
“write down the range”
“estimate the mean”
“find the standard deviation”
“find the variance”
“write down the mid-interval values”
“write down the modal class”

In disguise (3)

a stem-and-leaf diagram: it is a list, already in order, so count the leaves
percentages in place of frequencies: enter them as the frequencies; the mean, $\sigma$ and the median are the same as for the counts
"the average number of … per …" asked of a frequency table: the mean, $\frac{\sum fx}{n}$

Variants (2)

data: a short list (), a table of values with frequencies, or grouped data (with unequal widths, or with percentages as frequencies)
asked: the median, the quartiles, the IQR, the mode, the mean, the standard deviation, the variance, the modal class, the midpoints

Looks like it, but isn't (6)

a normal model with an interquartile range or a percentage given, and $\sigma$ asked: Unknown mean or standard deviation
values read off a box plot: Box plots
values read off a cumulative frequency graph: Cumulative frequency graphs
a mean given and a value missing: Work backwards from the mean or median
a table of probabilities, and $\mathrm{E}(X)$ asked: Probability distribution tables (the same sum, with probabilities in place of $\frac{f}{n}$ )
a frequency table, and the expected value for one randomly chosen or the next person (worded "expected" or "on average"): Probability distribution tables (the same sum, with the relative frequencies as the probabilities)

Worked example

A short list: order it, then count positions

Paper 1 (no calculator)

Question

The number of text messages sent in one day by each of $15$ people was recorded:

23, \ 42, \ 17, \ 28, \ 31, \ 14, \ 46, \ 23, \ 39, \ 22, \ 51, \ 26, \ 30, \ 19, \ 35

(a) Write down (i) the range (ii) the mode (iii) the median. [3]
(b) Find the interquartile range. [3]

Try it yourself first, then compare.

Now try one like it

Write your working on paper or on screen. Paper Zero marks it step by step, like the worked example, and shows where each mark went. Free, with no account.

Try one like it

Which method?

Given	Method
a short list, on	order it, then count to the positions
a table of values with frequencies, on	the running totals of the frequencies tell you which value is at each position
a table of values with frequencies, on	one-variable statistics with the frequencies as a second list
grouped data	midpoints with the frequencies: the mean and $\sigma$ are estimates
the standard deviation	technology only; the variance is $\sigma^2$ , as a written step

Steps

Organise the data.
Order a short list and count it. For a table, keep each value with its frequency. For grouped data, write the midpoints.
, positions:
the median is at $\frac{n + 1}{2}$ . Split the data into halves (for odd $n$ , leave the median out of both), and take the median of each half for $Q_1$ and $Q_3$ . For a table, count through the running totals of the frequencies. Write $\mathrm{IQR} = Q_3 - Q_1$ with each quartile labelled. The mean by hand is $\frac{\sum fx}{n}$ (booklet), with the midpoints as the $x$ -values for grouped data: write the sum with its terms.
, GDC:
one-variable statistics with the frequency list. Write each statistic by name with its value. The variance is $\sigma^2$ of the unrounded $\sigma$ .
Mode:
the value with the highest frequency (not the frequency). Modal class: the interval, with equal class widths.
Check:
$Q_1 \le$ median $\le Q_3$ , and the mean lies among the values where most of the data are. Give the accuracy the question asks for.

Common mistakes

Counting positions in an unordered list.
Order it first, and count that none are lost.
Giving the position as the median
"the 8th value" is where to look; the median is the value there.
Entering the values without their frequencies
on the GDC: in Example 1.2 that gives $2.5$ instead of $1.975$ .

More mistakes (6)

Reading $s$ instead of $\sigma$
$s$ is the sample value, and it is larger ( $1.37$ instead of $1.35$ in Example 1.2).
The variance
forgetting to square, or squaring a rounded $\sigma$ . Square the full value.
The mode as the highest frequency,
or the modal class as its midpoint: both score .
Halving the width instead of finding the midpoint
of an unequal class: $15 \le t < 25$ has midpoint $20$ .
Unlabelled quartiles
" $39 - 22$ " with no labels can lose the method mark if either number is wrong.
Ignoring the accuracy the question sets
"correct to one decimal place" (Example 1.3) or "to 4 significant figures" replaces the usual 3 s.f., and the final needs that form.

Easy to confuse with

Next typeCumulative frequency graphs

Now try one like it

Keep going

Easy to confuse with

More in Descriptive statistics: sampling, averages, spread and data displays