4.6.2 Histograms
A histogram is a graphical representation of a frequency distribution.
It is a type of bar chart that shows the distribution of data values in a set by dividing them into a set of intervals called bins or class intervals.
In a histogram,
The horizontal axis represents the range of values or intervals of the variable being measured
The vertical axis represents the frequency or density of data points falling within each interval
The bars of a histogram are drawn adjacent to each other with no gaps between them, to indicate that the variable is continuous.
Histograms are commonly used in statistics and data analysis to visualize the distribution of data, identify patterns and outliers, and explore the shape of the distribution.
They are especially useful for large data sets, where it can be difficult to see patterns and trends in the data when presented in a raw form.
Drawing Histogram:
To draw a histogram
Determine the range and number of intervals (or bins) for the variable you want to plot.
The number of bins will depend on the size of your data set and the nature of the data.
Count the frequency of data points falling into each interval.
This can be done manually by counting the number of data points in each interval, or by using software such as Excel or statistical packages like R, Python, or SPSS.
Calculate the frequency density for each interval by dividing the frequency by the width of the interval.
This will give you the height of the bars in your histogram.
Draw a horizontal axis that represents the range of values for the variable being measured, and divide it into intervals that match your data set.
Draw a vertical axis that represents the frequency or density of data points, and label it accordingly.
Draw a rectangle above each interval on the horizontal axis, with a height equal to the frequency density of that interval.
The bars should be adjacent to each other, with no gaps in between them.
Add a title and labels to your histogram, including the name of the variable being measured, the range of values on the horizontal axis, and the units of measurement.
Finally, check your histogram for accuracy and clarity, making sure that it accurately represents your data set and is easy to read and interpret.
Worked Example:
During one year the midday temperature, \( t°C \), in Zedford were recorded.
The table shows the results.
Complete the histogram to show the information in the table.
A histogram uses frequency density on \( y \)-axis, which is calculated by using
\( frequency \ density = \frac{frequency}{class \ width} \)
Calculate frequency density for each class width.
Complete the histogram, using class width as a width of bar and frequency density as a height of bar.
Test yourself
Question 1:
The speed, \( V \) km\h , of each of \( 200 \) cars passing a building is measured.
The frequency table shows the results.
On the grid, draw a histogram to show the information in this table.
[3]
Question 2:
Some students each record the mass, \( m \)kg, of their school bag.
Adil wants to draw a histogram to show this information.
Complete the table.
[2]
Question 3:
The height, \( h \) cm, of each of \( 120 \) plants is measured.
The frequency table shown this information.
A histogram is drawn to show the information in the frequency table.
The height of the bar representing the interval \( 10<h≤20 \) is \( 7.2 \)cm
Calculate the height of the bar representing the interval \( 30<h≤50 \).
[2]
Question 4:
The histogram shows information about the time taken by cyclists to finish a cycle race.
\( 7 \) cyclists took \( 80 \) minutes or less to finish the race.
Work out an estimate for the number of cyclists who took more than \( 105 \) minutes to finish the race.
[3]
Explain why your answer to part i is only an estimate.
[1]
Question 5:
The table gives information about the heights of \( 150 \) students.
On the grid, draw a histogram for this information.
[3]
Work out an estimate for the fraction of the students who have a height between \( 150 \) cm and \( 170 \) cm.
[2]