The table below shows information regarding the average speeds travelled by trains in a region of the UK.
The data is to be plotted on a histogram.
Work out the frequency density for each class interval.
Average speed s m/s |
Frequency |
5 | |
15 | |
28 | |
38 | |
14 |
Add two columns to the table - one for class width, one for frequency density.
Writing the calculation in each box helps to keep accuracy.
Average speed s m/s |
Frequency | Class width | Frequency density |
5 | 40 - 20 = 20 | 5 ÷ 20 = 0.25 | |
15 | 50 - 40 = 10 | 15 ÷ 10 = 1.5 | |
28 | 55 - 50 = 5 | 28 ÷ 5 = 5.6 | |
38 | 60 - 55 = 5 | 38 ÷ 5 = 7.6 | |
14 | 70 - 60 = 10 | 14 ÷ 10 = 1.4 |
A histogram is shown below representing the distances achieved by some athletes throwing a javelin.
There are two classes missing from the histogram. These are:
Distance, m | Frequency |
8 | |
2 |
Add these to the histogram.
Before completing the histogram, remember to show clearly you've worked out the missing frequency densities.
Distance, m | Frequency | Class width | Frequency density |
8 | 70 - 60 = 10 | 8 ÷ 10 = 0.8 | |
2 | 100 - 80 = 20 | 2 ÷ 20 = 0.1 |
The table below and its corresponding histogram show the mass, in kg, of some new born bottlenose dolphins.
Mass m kg |
Frequency |
4 ≤ m < 8 | 4 |
8 ≤ m < 10 | 15 |
10 ≤ m < 12 | 19 |
12 ≤ m < 15 | |
15 ≤ m < 30 | 6 |
Use the table and histogram to find the value of in the formula
Start by finding the frequency density in terms of - add two columns to the table, one for class width, one for frequency density.
Mass m kg |
Frequency | Class width | Frequency density |
4 ≤ m < 8 | 4 | 8 - 4 = 4 | |
8 ≤ m < 10 | 15 | 10 - 8 = 2 | |
10 ≤ m < 12 | 19 | 2 | |
12 ≤ m < 15 | 3 | ||
15 ≤ m < 30 | 6 | 15 |
We can use either of the two intervals that feature both in the table and on the histogram to find the value of .
Using the first bar.
Check using the other (2^{nd}) bar to check;
Estimate the number of dolphins whose weight is greater than 13 kg.
We can see from the table that their are 6 dolphins in the interval 15 ≤ m < 30.
So we need to estimate the number of dolphins that are in the interval 13 ≤ m < 15.
For 13 ≤ m < 15, the histogram shows the frequency density is 1.5 and we found the value of in part (a).
Using the formula given in the question,
So the total number of dolphins can be estimated by
There are approximately 12 dolphins with a weight greater than 13 kg