Cumulative Frequency Graphs
Mode, Mean and Median for Continuous Data
A sample of people were asked about their weights but many did not feel comfortable giving exact values so
an anonymous survey was conducted where they were asked to choose one of the following class intervals.
Here is the collected data:
| Weight ($x$ kg) in class intervals |
Frequency |
|---|---|
| $40 \le x \lt 50$ | $10$ |
| $50 \le x \lt 60$ | $18$ |
| $60 \le x \lt 70$ | $13$ |
| $70 \le x \lt 80$ | $15$ |
| $80 \le x \lt 90$ | $4$ |
How many people completed the survey?
Of the $4$ people in the class interval $80 \le x \lt 90$, how many weigh exactly $83$ kg?
0
1
2
3
4
cannot determine
1
2
3
4
cannot determine
Is it possible to determine any of the exact weights from the table?
Yes
No
Is it possible to determine the mode, mean or median from the table?
Yes
No
Mode
We cannot determine the mode but the modal class is
40 ≤ x < 50
50 ≤ x < 60
60 ≤ x < 70
70 ≤ x < 80
80 ≤ x < 90
50 ≤ x < 60
60 ≤ x < 70
70 ≤ x < 80
80 ≤ x < 90
Mean
We cannot determine the mean but we can estimate the mean. For example,
for the $10$ people in the interval $40 \le x \lt 50$, we can assign them each a weight of
for the $18$ people in the interval $50 \le x \lt 60$, we can assign them each a weight of
and so on.
for the $10$ people in the interval $40 \le x \lt 50$, we can assign them each a weight of
for the $18$ people in the interval $50 \le x \lt 60$, we can assign them each a weight of
and so on.
kg
kg
kg
| Weight ($x$ kg) in class intervals |
Mid-interval Value |
Frequency |
|---|---|---|
| $40 \le x \lt 50$ | $45$ | $10$ |
| $50 \le x \lt 60$ | $55$ | $18$ |
| $60 \le x \lt 70$ | $65$ | $13$ |
| $70 \le x \lt 80$ | $75$ | $15$ |
| $80 \le x \lt 90$ | $85$ | $4$ |
If the $10$ people in the interval $40 \le x \lt 50$ are each
assigned a weight of $45$ kg, then the total weight of these people is
assigned a weight of $45$ kg, then the total weight of these people is
kg
If we did this for the other class intervals and divided the total weight
by the total number of people, an estimate of the mean would be
by the total number of people, an estimate of the mean would be
kg
Median
We cannot determine the median but we know it will be in the interval
40 ≤ x < 50
50 ≤ x < 60
60 ≤ x < 70
70 ≤ x < 80
80 ≤ x < 90
50 ≤ x < 60
60 ≤ x < 70
70 ≤ x < 80
80 ≤ x < 90
To estimate the median, we use cumulative frequency.
How many people weigh less than 40 kg?
How many people weigh less than 50 kg?
How many people weigh less than 60 kg?
How many people weigh less than 70 kg?
How many people weigh less than 80 kg?
How many people weigh less than 90 kg?
| Weight ($x$ kg) in class intervals |
Frequency | Cumulative Frequency |
|---|---|---|
| $\phantom{30 \le}\; x \lt \color{red}{40}$ | $0$ | $\color{red}{0}$ |
| $40 \le x \lt \color{red}{50}$ | $10$ | $\color{red}{10}$ |
| $50 \le x \lt \color{red}{60}$ | $18$ | $\color{red}{28}$ |
| $60 \le x \lt \color{red}{70}$ | $13$ | $\color{red}{41}$ |
| $70 \le x \lt \color{red}{80}$ | $15$ | $\color{red}{56}$ |
| $80 \le x \lt \color{red}{90}$ | $4$ | $\color{red}{60}$ |
To draw a cumulative frequency curve, weights are shown on the horizontal axis and cumulative frequecy is on the vertical axis.
We plot points for the weights and cumulative frequencies and draw a smooth curve through them.
We plot points for the weights and cumulative frequencies and draw a smooth curve through them.
There are $60$ people so the median would be the average of the th and st person's weights.
However, the difference between the $30.5$th person and the $30$th person is small on the cumulative frequency graph so we use the $30$th person for simplicity.
From the cumulative frequency curve, an estimate of the median would be (to 1 decimal place) kg.
| Weight ($x$ kg) in class intervals |
Frequency | Cumulative Frequency |
|---|---|---|
| $\phantom{30 \le}\; x \lt \color{red}{40}$ | $0$ | $\color{red}{0}$ |
| $40 \le x \lt \color{red}{50}$ | $10$ | $\color{red}{10}$ |
| $50 \le x \lt \color{red}{60}$ | $18$ | $\color{red}{28}$ |
| $60 \le x \lt \color{red}{70}$ | $13$ | $\color{red}{41}$ |
| $70 \le x \lt \color{red}{80}$ | $15$ | $\color{red}{56}$ |
| $80 \le x \lt \color{red}{90}$ | $4$ | $\color{red}{60}$ |
Interquartile Range
Using this curve, we can also estimate the lower quartile ($Q_1$), upper quartile ($Q_3$) and hence the interquartile range.
If the median is the $30$th person's weight, the lower quartile ($Q_1$) is the th person's weight and the upper quartile ($Q_3$) is the th person's weight.
$Q_1=$
kg
$Q_3=$
kg
Interquartile Range $=$
kg
Estimate the number of people who weigh less than or equal to 56 kg.
people
Estimate the number of people who weigh more than or equal to 76 kg.
people
Percentiles
Find:
$50\%$ of $60=$
$60\%$ of $60=$
$90\%$ of $60=$
The 50th percentile is the weight which 50% of the people are less than. The 50th percentile =
kg
The 60th percentile is the weight which 60% of the people are less than. The 60th percentile =
kg
The 90th percentile is the weight which 90% of the people are less than. The 90th percentile =
kg