Measures of central tendency and dispersion

Mean, median and mode

Change the data and see where the mean (pink line) and the median (green line) end up:

Mean from frequency tables and intervals

When the data are in a table, each value counts according to its frequency: $\bar{x} = \dfrac{\sum x_i\, n_i}{N}$. If they are grouped in intervals, we use the class mark (midpoint) instead of $x_i$ and the result is an estimate.

When we group data, we lose information

Using class marks gives an estimate, not necessarily the exact mean. The specific values inside each interval are no longer known.

The range: the simplest measure of dispersion

The two phone models have the same mean battery life (15 h), but model A always lasts about the same: range $= 16 - 14 = 2$. Model B is very irregular: range $= 20 - 10 = 10$. For reliability, a small range is a good sign.

The mean deviation: how far the data are from the centre

The range only looks at the two extremes. The mean deviation takes all the data into account: how far, on average, each value is from the mean.

Investigation and spreadsheet: from data to decision

We want to study the minutes spent reading the previous day in two classes of 20 students. We pick 4 people at random from each class and ask the same question, for the same period and in the same unit. We record the answers without names and note any absences: we do not replace them with zero. Choosing at random reduces bias, but a sample of eight cannot guarantee a perfect representation of all 40 people; a single day does not describe all reading habits either.

Practise with these fictitious data: 20, 30, 30, 40, 40, 40, 50, 70 minutes.

1. In A1 write Minutes and in A2:A9, the data. In C1 write Measure and in D1, Result.

2. Complete C2:D4 with the first three rows of the table below.

3. In B1 write Distance from the mean; in B2, =ABS(A2-$D$2). Copy down to B9: the range should give 20, 10, 10, 0, 0, 0, 10, 30.

4. Complete C5:D6. To build a frequency table, write the values 20, 30, 40, 50, 60, 70 in F2:F7, and in G2, =CONTAR.SI($A$2:$A$9;F2). Copy down to G7: the frequencies are 1, 2, 3, 1, 0, 1.

5. Add headings in F1:G1 and create a column chart with separate columns, with "Minutes recorded" and "Students" on the axes. State in the title that there are eight responses and start the frequency axis at zero.

Use the equivalent functions if your application changes the names or separators. Here you are calculating with individual data: they are not estimates using class marks.

Reasoned conclusion: in the first data set, 40 minutes is the centre according to all three measures, but not everyone reads the same amount. We cannot conclude that every student in the school reads 40 minutes, or that one class reads more than another, without comparing their data separately. To guide a school activity, we would enlarge the sample and collect data over several days; we would not use these fictitious data as real evidence.

Cooperating, arguing and managing effort

In teams of three, rotate the roles of data collection, checking and explaining. Each person proposes one conclusion and one limitation; justify the final choice with data. If there is disagreement, try both proposals and listen before replying. Provide readable tables and enough time for the whole team to take part, without attributing abilities to gender or to who calculates fastest.

If a formula fails, name the difficulty: do you not understand the data, the range or the operation? Simplify the example, check it by hand and change one thing at a time. When you finish, explain which strategy helped you and which mistake taught you something new.

Summary