Analysing Frequency Tables
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Theory
A frequency table lists each value \(x\) once beside its frequency \(f\). Add an \(fx\) column and a cumulative frequency column, then \(\text{mean} = \dfrac{\sum fx}{\sum f}\) and the median is read from the running total. This Year 8 maths page shows how.
The columns. The value is \(x\) and its frequency is \(f\). The \(fx\) column is value \(\times\) frequency, and the cumulative frequency is a running total of \(f\). The symbol \(\sum\) means “add up”.
How many values: \(\sum f\), the sum of the frequency column; never add the value column. Range: largest \(x\) minus smallest \(x\). Mode: the value \(x\) in the row with the largest \(f\).
Mean: each row contributes \(fx\) to the total, so \(\text{mean} = \dfrac{\sum fx}{\sum f}\).
Median: find the middle position(s) from \(\sum f\), then read down the cumulative frequency column to the first row that reaches each position.
Relative frequency of a row \(= \dfrac{f}{\sum f}\); multiply by \(100\) for a percentage. The relative frequencies add to \(1\).
A missing frequency can be found from the total: the frequencies add to \(\sum f\).
Grouped data. The class centre is halfway between the ends of a class. The modal class has the largest \(f\). Using the class centres as \(x\) gives only an estimate of the mean.
The four columns for the nights of sport:
| x | f | fx | Cumulative |
|---|---|---|---|
| 0 | 4 | 0 | 4 |
| 1 | 8 | 8 | 12 |
| 2 | 10 | 20 | 22 |
| 3 | 6 | 18 | 28 |
| 4 | 2 | 8 | 30 |
| Total | 30 | 54 |
Statistics from a frequency table
- Add the \(f\) column for the number of values.
- Build the \(fx\) column and add it for the total; mean \(= \sum fx \div \sum f\).
- Build the cumulative frequency column; read down to the middle position(s) for the median.
- Read the mode (largest \(f\)) and the range (largest \(x\) \(-\) smallest \(x\)).
Grouped data
- Find each class centre.
- Multiply centre \(\times\) frequency, add, and divide by \(\sum f\) for an estimated mean.
(i) the range
(ii) the mode
(iii) the median
(iv) the mean.
There are \(16\) scores, so the median is halfway between the 8th and 9th. The cumulative frequencies are \(1, 3, 6, 10, \ldots\), so both are \(7\).
| \(\text{Range}\) | \(=\) | \(10 - 4 = 6\) |
| \(\text{Median}\) | \(=\) | \(\dfrac{7 + 7}{2} = 7\) |
| \(\sum fx\) | \(=\) | \(4 + 10 + 18 + 28\) |
| \(\) | \(\) | \(+\ 24 + 18 + 10\) |
| \(\) | \(=\) | \(112\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{112}{16} = 7\) |
(i) \(6\)
(ii) \(7\) (frequency \(4\))
(iii) \(7\)
(iv) \(7\)
| Score x | f | fx |
|---|---|---|
| 5 | 2 | 10 |
| 6 | 4 | 24 |
| 7 | 3 | 21 |
| 8 | 5 | 40 |
| 9 | 2 | 18 |
| Total | 16 | 113 |
| \(\bar{x}\) | \(=\) | \(\dfrac{113}{16}\) |
| \(\) | \(\approx\) | \(7.1\) |
The mean is about \(7.1\).
| \(\sum fx\) | \(=\) | \(2 + 10 + 18 + 16 + 15\) |
| \(\) | \(=\) | \(61\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{61}{20} = 3.05\) |
\(3.05\) debates per member.
(i) Find the median.
(ii) What percentage packed lunch on more than \(3\) days?
Cumulative frequencies: \(3, 7, 12, 18, 22, 24\). The 12th value is \(2\) and the 13th is \(3\).
| \(\text{Median}\) | \(=\) | \(\dfrac{2 + 3}{2} = 2.5\) |
| \(\dfrac{4 + 2}{24}\) | \(=\) | \(0.25 = 25\%\) |
(i) \(2.5\) days
(ii) \(25\%\)
(i) Find \(k\).
(ii) Find the mean.
(iii) Find the median.
| \(5 + k + 6 + 3 + 1\) | \(=\) | \(25\) |
| \(k\) | \(=\) | \(10\) |
| \(5 + 20 + 18\) | \(=\) | \(43\) |
| \(43 + 12 + 5\) | \(=\) | \(60\) |
| \(\text{Mean}\) | \(=\) | \(\dfrac{60}{25} = 2.4\) |
(i) \(k = 10\)
(ii) \(2.4\)
(iii) The 13th value: running totals \(5, 15\) put the 6th to 15th at \(2\), so the median is \(2\).
(i) Find the class centre of \(26\)–\(30\).
(ii) Name the modal class.
(iii) Estimate the mean.
| \(\dfrac{26 + 30}{2}\) | \(=\) | \(28\) |
| \(\sum fx\) | \(=\) | \(72 + 253 + 280 + 165\) |
| \(\) | \(=\) | \(770\) |
| \(\text{Mean}\) | \(\approx\) | \(770 \div 30\) |
| \(\) | \(\approx\) | \(25.7\) |
(i) \(28\)
(ii) \(21\)–\(25\) minutes
(iii) About \(25.7\) minutes; only an estimate, since every runner in a class counts as its centre.
Common pitfalls
Frequently asked questions
How do you find the mean from a frequency table?
Multiply each value by its frequency (\(fx\)), add these, then divide by the total frequency.
What is cumulative frequency?
A running total of the frequencies, row by row.
How do you find the median from a frequency table?
Find the middle position from the total frequency, then go down the cumulative frequency column to the first row that reaches it.
What is relative frequency?
A row’s frequency divided by the total frequency: \(8\) out of \(40\) is \(0.2\).
What is a class centre?
The number halfway between the ends of a class: for \(10\)–\(14\) it is \(12\).