Grouped Data
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- Grouped Data - Video - Mean, Median, and Mode of Grouped Data & Frequency Distribution Tables Statistics Watch
Theory
A histogram shows how often each score, or each group of scores, occurs. This Year 9 maths page shows how to read and draw histograms, group data into class intervals, find the modal class and the class centres, draw a frequency polygon, find the mean and median from a histogram, and estimate the mean when only the classes are known.
A histogram shows numerical data. The scores run along the horizontal axis and the frequency (how many times each score occurs) runs up the vertical axis. The columns touch, because the horizontal axis is a number line. For single scores, each column is centred on its score.
A column graph shows categories, such as favourite sport, so its columns have gaps.
For data with many different values, the scores are grouped into class intervals of equal width. The class \(50\)–\(55\) usually means from \(50\) up to, but not including, \(55\), so \(55\) belongs to the next class. For whole numbers the classes may be written \(0\)–\(9\), \(10\)–\(19\) and so on.
The class centre is the middle of a class. The modal class is the class with the highest frequency, the tallest column.
A frequency polygon joins the midpoints of the tops of the columns with straight lines. It is closed off on the horizontal axis half a class width beyond each end.
Once data are grouped, the exact scores are lost. The histogram still gives totals and percentages for whole classes, but values inside a class can only be estimated.
For the class \(60\)–\(65\) cm the centre is \(62.5\) cm. For whole-number classes such as \(10\)–\(19\), the centre is \(\dfrac{10 + 19}{2} = 14.5\).
How to work with a histogram
- Read the axes. Are the columns single scores or classes? What does a class such as \(50\)–\(55\) include?
- Count: add the column heights.
- Mode: the score or class under the tallest column.
- Mean: multiply each score, or each class centre, by its frequency, add, and divide by the total. With classes, call it an estimate.
- Median: add the heights from the left to position \(\dfrac{n+1}{2}\).
- Drawing: use an even scale on both axes, make the columns touch, and label both axes. For a frequency polygon, mark the midpoint of each column top, join them, and close off at the class centres either side.
The tallest column is above \(1\), so the mode is \(1\) goal. For the mean, multiply each score by its frequency.
\(\sum f = 3 + 7 + 6 + 3 + 1 = 20\)
\(\sum fx = 0 + 7 + 12 + 9 + 4 = 32\)
| \(\text{mean}\) | \(=\) | \(\dfrac{32}{20} = \mathbf{1.6}\) |
With \(20\) scores the median is between the \(10\)th and \(11\)th. The \(0\)s fill places \(1\)–\(3\) and the \(1\)s places \(4\)–\(10\), so the \(10\)th is \(1\) and the \(11\)th is \(2\).
| \(\text{median}\) | \(=\) | \(\dfrac{1 + 2}{2} = \mathbf{1.5}\) |
| Height (cm) | \(10\)–\(15\) | \(15\)–\(20\) | \(20\)–\(25\) | \(25\)–\(30\) | \(30\)–\(35\) |
|---|---|---|---|---|---|
| Frequency | \(3\) | \(8\) | \(12\) | \(5\) | \(2\) |
(ii) Where does a frequency polygon meet the horizontal axis?
(iii) What percentage of the seedlings are \(25\) cm or taller?
(i) The highest frequency is \(12\), so the modal class is \(20\)–\(25\) cm.
(ii) The polygon is closed off half a class width (\(2.5\) cm) beyond each end, at the centres of the empty classes \(5\)–\(10\) and \(35\)–\(40\): at \(7.5\) cm and \(37.5\) cm.
(iii) Heights of \(25\) cm or more are in the last two classes.
| \(5 + 2\) | \(=\) | \(7\) |
| \(\dfrac{7}{30} \times 100\%\) | \(\approx\) | \(\mathbf{23.3\%}\) |
Replace every height by its class centre, then multiply by the frequency.
| Class | Centre \(c\) | \(f\) | \(f \times c\) |
|---|---|---|---|
| \(10\)–\(15\) | \(12.5\) | \(3\) | \(37.5\) |
| \(15\)–\(20\) | \(17.5\) | \(8\) | \(140\) |
| \(20\)–\(25\) | \(22.5\) | \(12\) | \(270\) |
| \(25\)–\(30\) | \(27.5\) | \(5\) | \(137.5\) |
| \(30\)–\(35\) | \(32.5\) | \(2\) | \(65\) |
| \(\sum (f \times \text{centre})\) | \(=\) | \(650\) |
| \(\text{estimated mean}\) | \(=\) | \(\dfrac{650}{30}\) |
| \(\) | \(\approx\) | \(\mathbf{21.67}\text{ cm}\) |
The median is between the \(15\)th and \(16\)th heights. Running totals are \(3\), \(11\), \(23\), so both are in the class \(20\)–\(25\) cm.
It is an estimate because the exact heights inside each class are not known. Each one has been replaced by its class centre.
Smallest: put every mark at the bottom of its class. Largest: put every mark at the top.
Smallest total: \(2 \times 40\) \(+ \, 5 \times 50\) \(+ \, 8 \times 60\) \(+ \, 5 \times 70\)
Largest total: \(2 \times 49\) \(+ \, 5 \times 59\) \(+ \, 8 \times 69\) \(+ \, 5 \times 79\)
| \(\text{smallest}\) | \(=\) | \(80 + 250 + 480\) |
| \(\) | \(\) | \({} + 350 = 1160\) |
| \(\text{largest}\) | \(=\) | \(98 + 295 + 552\) |
| \(\) | \(\) | \({} + 395 = 1340\) |
| \(\text{smallest mean}\) | \(=\) | \(\dfrac{1160}{20} = \mathbf{58}\) |
| \(\text{largest mean}\) | \(=\) | \(\dfrac{1340}{20} = \mathbf{67}\) |
Every mark moved up by \(9\), so the mean moved up by \(9\) too. The true mean is somewhere from \(58\) to \(67\).
Common pitfalls
Frequently asked questions
What is the difference between a histogram and a column graph?
A histogram shows numerical data on a number line, so its columns touch. A column graph shows categories, such as favourite sports, so its columns have gaps.
What is a class centre?
The middle of a class interval: add the two ends and divide by \(2\). The class \(60\)–\(65\) has a centre of \(62.5\).
What is the modal class?
The class with the highest frequency, shown by the tallest column of a grouped histogram.
How do you draw a frequency polygon?
Mark a point at the middle of the top of each column, join the points with straight lines, and close the polygon on the horizontal axis half a class width beyond each end.
How do you estimate the mean of grouped data?
Multiply each class centre by its frequency, add the results, and divide by the total frequency. It is only an estimate because the exact values in each class are unknown.