Measure of Variation
6.1 Measure of Variation
Measures of centre of a data set
These measures include mean, median, and mode.
For a set of data, the mean is defined by the sum of the values, divided by the number of values.
The median is the middle point of the data arranged in ascending order.
The mode is the value that occurs most often in a set of data.
The simplest measure of spread is the range R which is the difference between the largest value and the smallest value.
The data variation is on the difference of each data value from the mean. This difference is called a deviation, in symbol $x_i - \bar{x}$.
The variance $\sigma^2$ is the mean of squares of the deviations.
The standard deviation $\sigma$ is the square root of the variance.
We can compute the similar formula for the variance as follow:
Quartiles $Q_1, Q_2$ and $Q_3$ divide the data arranged in ascending order into four equal groups. Here, $Q_2$ is the median of the whole data set. $Q_1$ and $Q_3$ are called the lower quartile and the upper quartile. The interquartile range is a measure of variability.
From the frequency table, we can find the mean:
We have the similar formula for the variance of group data as follow:
Examples
Example 1
The marks scored in a test by seven students are 3, 4, 5, 2, 8, 8, 5. Find the mean and standard deviation.
Step 1: Calculate the Mean ($\bar{x}$)
$ \bar{x} = \frac{3+4+5+2+8+8+5}{7} = \frac{35}{7} = 5 $
Step 2: Calculate the Variance ($\sigma^2$) using Computational Formula
$\sum x^2 = 3^2+4^2+5^2+2^2+8^2+8^2+5^2 = 9+16+25+4+64+64+25 = 207$
$\sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2 = \frac{207}{7} - (5)^2 = 29.5714 - 25 = 4.5714$
Step 3: Calculate the Standard Deviation ($\sigma$)
$ \sigma = \sqrt{4.5714} = 2.1381 $
Example 2
Ten students are weighted (w kg). The summary data for the weights are $\sum w = 440$, $\sum w^2 = 19490$. Find the mean and standard deviation of the students' weight.
Step 1: Calculate the Mean ($\bar{w}$)
$ \bar{w} = \frac{\sum w}{n} = \frac{440}{10} = 44 $
Step 2: Calculate the Variance ($\sigma^2$)
$ \sigma^2 = \frac{\sum w^2}{n} - (\bar{w})^2 = \frac{19490}{10} - (44)^2 = 1949 - 1936 = 13 $
Step 3: Calculate the Standard Deviation ($\sigma$)
$ \sigma = \sqrt{13} = 3.6056 $
Example 3
Find the median and interquartile range for each of the set of data below.
(a) 7, 9, 4, 6, 3, 2, 8, 1, 10, 15, 11
Step 1: Arrange data in ascending order.
1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 15
Step 2: Find the Median ($Q_2$).
Median ($Q_2$) = 7 (the 6th value)
Step 3: Find the Lower Quartile ($Q_1$) and Upper Quartile ($Q_3$).
Lower half: [1, 2, 3, 4, 6]. $Q_1 = 3$.
Upper half: [8, 9, 10, 11, 15]. $Q_3 = 10$.
Step 4: Calculate the Interquartile Range (IQR).
IQR = $Q_3 - Q_1 = 10 - 3 = 7$.
(b) 6, 5, 1, 14, 3, 7, 8, 2, 13
Step 1: Arrange data in ascending order.
1, 2, 3, 5, 6, 7, 8, 13, 14
Step 2: Find the Median ($Q_2$).
Median ($Q_2$) = 6 (the 5th value)
Step 3: Find $Q_1$ and $Q_3$.
Lower half: [1, 2, 3, 5]. $Q_1 = \frac{2+3}{2} = 2.5$.
Upper half: [7, 8, 13, 14]. $Q_3 = \frac{8+13}{2} = 10.5$.
Step 4: Calculate the IQR.
IQR = $Q_3 - Q_1 = 10.5 - 2.5 = 8$.
(c) 2, 15, 13, 6, 5, 12, 50, 22, 18, 52
Step 1: Arrange data in ascending order.
2, 5, 6, 12, 13, 15, 18, 22, 50, 52
Step 2: Find the Median ($Q_2$).
Median ($Q_2$) = $\frac{13+15}{2} = 14$.
Step 3: Find $Q_1$ and $Q_3$.
Lower half: [2, 5, 6, 12, 13]. $Q_1 = 6$.
Upper half: [15, 18, 22, 50, 52]. $Q_3 = 22$.
Step 4: Calculate the IQR.
IQR = $Q_3 - Q_1 = 22 - 6 = 16$.
(d) 5, 3, 5, 7, 8, 2, 4, 1, 2, 2, 3, 6
Step 1: Arrange data in ascending order.
1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8
Step 2: Find the Median ($Q_2$).
Median ($Q_2$) = $\frac{3+4}{2} = 3.5$.
Step 3: Find $Q_1$ and $Q_3$.
Lower half: [1, 2, 2, 2, 3, 3]. $Q_1 = \frac{2+2}{2} = 2$.
Upper half: [4, 5, 5, 6, 7, 8]. $Q_3 = \frac{5+6}{2} = 5.5$.
Step 4: Calculate the IQR.
IQR = $Q_3 - Q_1 = 5.5 - 2 = 3.5$.
Example 4
Find the mean and standard deviation for the following frequency table:
| x | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|
| Frequency(f) | 9 | 9 | 10 | 12 | 10 |
Step 1: Construct the calculation table
| $x$ | $f$ | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|
| 5 | 9 | 45 | 25 | 225 |
| 6 | 9 | 54 | 36 | 324 |
| 7 | 10 | 70 | 49 | 490 |
| 8 | 12 | 96 | 64 | 768 |
| 9 | 10 | 90 | 81 | 810 |
| $\sum f = 50$ | $\sum fx = 355$ | $\sum fx^2 = 2617$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{\sum fx}{\sum f} = \frac{355}{50} = 7.1000$
Variance: $\sigma^2 = \frac{\sum fx^2}{\sum f} - (\bar{x})^2 = \frac{2617}{50} - (7.1)^2 = 52.34 - 50.41 = 1.9300$
Standard Deviation: $\sigma = \sqrt{1.93} = 1.3892$
Example 5
A man recorded the length, in minutes, of each telephone call he made for a month. Find the mean and standard deviation for the data.
| Length (minute) | (0, 10] | (10, 20] | (20, 30] | (30, 40] | (40, 50] |
|---|---|---|---|---|---|
| Frequency (f) | 4 | 15 | 5 | 12 | 10 |
Step 1: Construct the calculation table with midpoints
| Length | Midpoint (x) | f | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|---|
| (0, 10] | 5 | 4 | 20 | 25 | 100 |
| (10, 20] | 15 | 15 | 225 | 225 | 3375 |
| (20, 30] | 25 | 5 | 125 | 625 | 3125 |
| (30, 40] | 35 | 12 | 420 | 1225 | 14700 |
| (40, 50] | 45 | 10 | 450 | 2025 | 20250 |
| $\sum f = 46$ | $\sum fx = 1240$ | $\sum fx^2 = 41150$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{\sum fx}{\sum f} = \frac{1240}{46} = 26.9565$
Variance: $\sigma^2 = \frac{\sum fx^2}{\sum f} - (\bar{x})^2 = \frac{41150}{46} - (\frac{1240}{46})^2 = 176.6068$
Standard Deviation: $\sigma = \sqrt{176.6068} = 13.2893$
Exercises 6.1
1. Find the median and the interquartile range for the following sets of data:
(a) 16, 18, 22, 19, 3, 21, 17, 20
Sorted Data: 3, 16, 17, 18, 19, 20, 21, 22
Median ($Q_2$) = $\frac{18+19}{2} = 18.5$.
Lower half: [3, 16, 17, 18]. $Q_1 = \frac{16+17}{2} = 16.5000$.
Upper half: [19, 20, 21, 22]. $Q_3 = \frac{20+21}{2} = 20.5$.
IQR = $20.5 - 16.5 = 4$.
(b) 33, 38, 43, 30, 29, 40, 51
Sorted Data: 29, 30, 33, 38, 40, 43, 51
Median ($Q_2$) = 38 (the 4th value).
Lower half: [29, 30, 33]. $Q_1 = 30$.
Upper half: [40, 43, 51]. $Q_3 = 43$.
IQR = $43 - 30 = 13$.
(c) 24, 32, 54, 31, 16, 18, 19, 14, 17, 20
Sorted Data: 14, 16, 17, 18, 19, 20, 24, 31, 32, 54
Median ($Q_2$) = $\frac{19+20}{2} = 19.5$.
Lower half: [14, 16, 17, 18, 19]. $Q_1 = 17$.
Upper half: [20, 24, 31, 32, 54]. $Q_3 = 31$.
IQR = $31 - 17 = 14$.
(d) 14, 16, 27, 18, 13, 19, 36, 15, 20
Sorted Data: 13, 14, 15, 16, 18, 19, 20, 27, 36
Median ($Q_2$) = 18 (the 5th value).
Lower half: [13, 14, 15, 16]. $Q_1 = \frac{14+15}{2} = 14.5$.
Upper half: [19, 20, 27, 36]. $Q_3 = \frac{20+27}{2} = 23.5$.
IQR = $23.5 - 14.5 = 9$.
2. Fifteen students do a mathematics test. Their marks are as follows: 7, 4, 9, 7, 6, 10, 12, 11, 3, 8, 5, 9, 8, 7, 3. Find the median and the interquartile range.
Step 1: Arrange data in ascending order.
3, 3, 4, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 11, 12
Step 2: Find the Median ($Q_2$).
Median ($Q_2$) = 7 (the 8th value)
Step 3: Find $Q_1$ and $Q_3$.
Lower half: [3, 3, 4, 5, 6, 7, 7]. $Q_1 = 5$.
Upper half: [8, 8, 9, 9, 10, 11, 12]. $Q_3 = 9$.
Step 4: Calculate the IQR.
IQR = $Q_3 - Q_1 = 9 - 5 = 4$.
3. Find the mean, variance and the standard deviation of the following data sets.
(a) 1, 2, 3, 4, 5, 6, 7
Mean: $\bar{x} = \frac{1+2+3+4+5+6+7}{7} = \frac{28}{7} = 4.0000$
$\sum x^2 = 1^2+2^2+3^2+4^2+5^2+6^2+7^2 = 1+4+9+16+25+36+49=140$
Variance: $\sigma^2 = \frac{140}{7} - (4)^2 = 20 - 16 = 4$
Standard Deviation: $\sigma = \sqrt{4} = 2$
(b) 5, 6, 7, 3, 2, 9, 11
Mean: $\bar{x} = \frac{5+6+7+3+2+9+11}{7} = \frac{43}{7} \approx 6.1429$
$\sum x^2 = 5^2+6^2+7^2+3^2+2^2+9^2+11^2 = 25+36+49+9+4+81+121=325$
Variance: $\sigma^2 = \frac{325}{7} - (\frac{43}{7})^2 \approx 46.4286 - 37.7347 = 8.6939$
Standard Deviation: $\sigma = \sqrt{8.6939} = 2.9485$
(c) 4, 3, 2, 7, 0, 9
Mean: $\bar{x} = \frac{4+3+2+7+0+9}{6} = \frac{25}{6} \approx 4.1667$
$\sum x^2 = 4^2+3^2+2^2+7^2+0^2+9^2 = 16+9+4+49+0+81=159$
Variance: $\sigma^2 = \frac{159}{6} - (\frac{25}{6})^2 \approx 26.5 - 17.3611 = 9.1389$
Standard Deviation: $\sigma = \sqrt{9.1389} = 3.0231$
(d) 5, 5, 5, 4, 2, 7, 7, 7, 7
Mean: $\bar{x} = \frac{5(3)+4+2+7(4)}{9} = \frac{15+4+2+28}{9} = \frac{49}{9} \approx 5.4444$
$\sum x^2 = 5^2(3)+4^2+2^2+7^2(4) = 25(3)+16+4+49(4) = 75+16+4+196 = 291$
Variance: $\sigma^2 = \frac{291}{9} - (\frac{49}{9})^2 \approx 32.3333 - 29.6420 = 2.6913$
Standard Deviation: $\sigma = \sqrt{2.6913} = 1.6405$
(e) 1, 3, 5, 5, 7, 9, 10, 15, 17
Mean: $\bar{x} = \frac{1+3+5+5+7+9+10+15+17}{9} = \frac{72}{9} = 8$
$\sum x^2 = 1^2+3^2+5^2+5^2+7^2+9^2+10^2+15^2+17^2 = 1+9+25+25+49+81+100+225+289=804$
Variance: $\sigma^2 = \frac{804}{9} - (8)^2 = 89.3333 - 64 = 25.3333$
Standard Deviation: $\sigma = \sqrt{25.3333} = 5.0332$
4. The numbers of errors, x, on each of 200 pages of typescript was monitored. The results when summarized showed that $\sum x = 1000, \sum x^2 = 5500$. Calculate the mean and the standard deviation of the number of errors per page.
Step 1: Calculate the Mean ($\bar{x}$)
$\bar{x} = \frac{\sum x}{n} = \frac{1000}{200} = 5$
Step 2: Calculate the Variance ($\sigma^2$)
$\sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2 = \frac{5500}{200} - (5)^2 = 27.5 - 25 = 2.5$
Step 3: Calculate the Standard Deviation ($\sigma$)
$\sigma = \sqrt{2.5} = 1.5811$
5. In a student group, a record was kept for the number of absent days each student had over one particular term. Calculate the mean and standard deviation for these data shown in the table below.
| Number of absent days (x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Number of students (f) | 15 | 10 | 20 | 8 | 2 |
Step 1: Construct the calculation table
| $x$ | $f$ | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|
| 0 | 15 | 0 | 0 | 0 |
| 1 | 10 | 10 | 1 | 10 |
| 2 | 20 | 40 | 4 | 80 |
| 3 | 8 | 24 | 9 | 72 |
| 4 | 2 | 8 | 16 | 32 |
| $\sum f = 55$ | $\sum fx = 82$ | $\sum fx^2 = 194$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{\sum fx}{\sum f} = \frac{82}{55} = 1.4909$
Variance: $\sigma^2 = \frac{\sum fx^2}{\sum f} - (\bar{x})^2 = \frac{194}{55} - (1.4909)^2 = 3.5273 - 2.2228 = 1.3045$
Standard Deviation: $\sigma = \sqrt{1.3045} = 1.1421$
6. A moth trap was set every night for five weeks. The number of moths caught in the trap was recorded. Calculate the mean and standard deviation for these data shown in the table below.
| Number of moths (x) | 7 | 8 | 9 | 10 | 11 |
|---|---|---|---|---|---|
| Frequency (f) | 3 | 8 | 9 | 5 | 9 |
Step 1: Construct the calculation table
| $x$ | $f$ | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|
| 7 | 3 | 21 | 49 | 147 |
| 8 | 8 | 64 | 64 | 512 |
| 9 | 9 | 81 | 81 | 729 |
| 10 | 5 | 50 | 100 | 500 |
| 11 | 9 | 99 | 121 | 1089 |
| $ \sum f = 34$ | $\sum fx = 315$ | $\sum fx^2=2977$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{315}{34} = 9.2647$
Variance: $\sigma^2 = \frac{2977}{34} - (9.2647)^2 = 87.5588 - 85.8259 = 1.7329$
Standard Deviation: $\sigma = \sqrt{1.7329} = 1.3164$
7. The lifetimes of 80 batteries, to the nearest hour (h), is shown in the following table. Calculate the mean and standard deviation for these data.
| Lifetime (h) | (6, 10] | (10, 15] | (15, 20] | (20, 25] | (25, 30] |
|---|---|---|---|---|---|
| Number of batteries | 12 | 8 | 15 | 30 | 15 |
Step 1: Construct the calculation table
| Midpoint(x) | f | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|
| 8 | 12 | 96 | 64 | 768 |
| 12.5 | 8 | 100 | 156.25 | 1250 |
| 17.5 | 15 | 262.5 | 306.25 | 4593.75 |
| 22.5 | 30 | 675 | 506.25 | 15187.5 |
| 27.5 | 15 | 412.5 | 756.25 | 11343.75 |
| $\sum f=80$ | $\sum fx=1546$ | $\sum fx^2=33143$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{1546}{80} = 19.3250$
Variance: $\sigma^2 = \frac{33143}{80} - (19.3250)^2 = 414.2875 - 373.4556 = 40.8319$
Standard Deviation: $\sigma = \sqrt{40.8319} = 6.39$
8. The table summarizes the distances travelled by 150 students to college each day. Calculate the mean and standard deviation for these data.
| Distance (s) | (0, 2] | (2, 4] | (4, 6] | (6, 8] | (8, 10] | (10, 12] |
|---|---|---|---|---|---|---|
| Number of students | 8 | 22 | 55 | 35 | 18 | 12 |
Step 1: Construct the calculation table
| Distance (s) | Midpoint(x) | f | $f \cdot x$ | $x^2$ | $f \cdot x^2$ |
|---|---|---|---|---|---|
| (0, 2] | 1 | 8 | 8 | 1 | 8 |
| (2, 4] | 3 | 22 | 66 | 9 | 198 |
| (4, 6] | 5 | 55 | 275 | 25 | 1375 |
| (6, 8] | 7 | 35 | 245 | 49 | 1715 |
| (8, 10] | 9 | 18 | 162 | 81 | 1458 |
| (10, 12] | 11 | 12 | 132 | 121 | 1452 |
| $\sum f=150$ | $\sum fx=888$ | $\sum fx^2=6206$ |
Step 2: Calculate Mean, Variance, and Standard Deviation
Mean: $\bar{x} = \frac{\sum fx}{\sum f} = \frac{888}{150} = 5.92$
Variance: $\sigma^2 = \frac{\sum fx^2}{\sum f} - (\bar{x})^2 = \frac{6206}{150} - (5.92)^2 = 41.3733 - 35.0464 = 6.3269$
Standard Deviation: $\sigma = \sqrt{6.3269} = 2.5153$