Histograms
Histograms
A histogram is a graphical representation of how many times different, mutually exclusive events are observed in an experiment. To interpret a histogram, we find the events on the \(x\)-axis and the counts on the \(y\)-axis. Each event has a rectangle that shows what its count (or frequency) is.
Example
Question
Use the following histogram to determine the events that were recorded and the relative frequency of each event. Summarise your answer in a table.
Determine the events
The events are shown on the \(x\)-axis. In this example we have “not yet in school”, “in primary school” and “in high school”.
Read off the count for each event
The counts are shown on the \(y\)-axis and the height of each rectangle shows the frequency for each event.
- not yet in school: \(\text{2}\)
- in primary school: \(\text{5}\)
- in high school: \(\text{9}\)
Calculate relative frequency
The relative frequency of an event in an experiment is the number of times that the event occurred divided by the total number of times that the experiment was completed. In this example we add up the frequencies for all the events to get a total frequency of \(\text{16}\). Therefore the relative frequencies are:
- not yet in school: \(\cfrac{2}{16} = \cfrac{1}{8}\)
- in primary school: \(\cfrac{5}{16}\)
- in high school: \(\cfrac{9}{16}\)
Summarise
| Event | Count | Relative frequency |
| not yet in school | \(\text{2}\) | \(\cfrac{1}{8}\) |
| in primary school | \(\text{5}\) | \(\cfrac{5}{16}\) |
| in high school | \(\text{9}\) | \(\cfrac{9}{16}\) |
To draw a histogram of a data set containing numbers, the numbers first have to be grouped. Each group is defined by an interval. We then count how many times numbers from each group appear in the data set and draw a histogram using the counts.
Example
Question
The following data represent the heights of \(\text{16}\) adults in centimetres.
\[\begin{array}{l}\text{162}\ ;\ \text{168}\ ;\ \text{177}\ ;\ \text{147}\ ;\ \text{189}\ ;\ \text{171}\ ;\ \text{173}\ ;\ \text{168} \\\text{178}\ ;\ \text{184}\ ;\ \text{165}\ ;\ \text{173}\ ;\ \text{179}\ ;\ \text{166}\ ;\ \text{168}\ ;\ \text{165}\end{array}\]Divide the data into \(\text{5}\) equal length intervals between \(\text{140}\) \(\text{cm}\) and \(\text{190}\) \(\text{cm}\) and draw a histogram.
Determine intervals
To have \(\text{5}\) intervals of the same length between \(\text{140}\) and \(\text{190}\), we need and interval length of \(\text{10}\).Therefore the intervals are \((\text{140};\text{150}]\); \((\text{150};\text{160}]\); \((\text{160};\text{170}]\); \((\text{170};\text{180}]\); and \((\text{180};\text{190}]\).
Count data
The following table summarises the number of data values in each of the intervals.
| Interval | \((\text{140};\text{150}]\) | \((\text{150};\text{160}]\) | \((\text{160};\text{170}]\) | \((\text{170};\text{180}]\) | \((\text{180};\text{190}]\) |
| Count | \(\text{1}\) | \(\text{0}\) | \(\text{7}\) | \(\text{6}\) | \(\text{2}\) |
Draw the histogram
This lesson is part of:
Statistics and Probability