Practise mean, variance, standard deviation, coefficient of variation, quartiles and box plot. This page helps you understand how data are spread from minimum to maximum.
A percentile is a value below which a certain percentage of observations lie. The 25th percentile is \(Q_1\), the 50th percentile is the median \(Q_2\), and the 75th percentile is \(Q_3\).
To find quartiles by the percentile technique, arrange the data in ascending order and use the percentile position. If the position is an integer, take the average of that position and the next position.
A box plot is made from the five number summary: minimum, \(Q_1\), median, \(Q_3\), and maximum.
\[ i=\frac{p}{100}\times n \]
\[ \bar{x}=\frac{\sum x}{n} \]
\[ s^2=\frac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1} \]
\[ CV=\frac{s}{\bar{x}}\times 100\% \]
Monthly starting salaries for a sample of 12 business school graduates are given below.
| Graduate | Salary ($) | Graduate | Salary ($) |
|---|---|---|---|
| 1 | 3450 | 7 | 3490 |
| 2 | 3550 | 8 | 3730 |
| 3 | 3650 | 9 | 3540 |
| 4 | 3480 | 10 | 3925 |
| 5 | 3355 | 11 | 3520 |
| 6 | 3310 | 12 | 3480 |
Calculate the mean, standard deviation, variance and coefficient of variation. Then calculate the three quartiles and identify which quartile represents the median.
Arrange the data in ascending order and try at least one answer first. Even a wrong attempt will help you remember the method.
Given salary data:
\(3450,\;3550,\;3650,\;3480,\;3355,\;3310,\;3490,\;3730,\;3540,\;3925,\;3520,\;3480\)
Sample mean:
\[ \bar{x}=\frac{\sum x}{n} \]
\[ \bar{x}=\frac{42480}{12} \]
\[ \bar{x}=3540.00 \]
Sample variance:
\[ s^2=\frac{\sum x^2-\frac{(\sum x)^2}{n}}{n-1} \]
\[ s^2=\frac{150681050-\frac{42480^2}{12}}{11} \]
\[ s^2=27440.91 \]
Sample standard deviation:
\[ s=\sqrt{s^2} \]
\[ s=\sqrt{27440.91} \]
\[ s=165.65 \]
Coefficient of variation:
\[ CV=\frac{s}{\bar{x}}\times 100 \]
\[ CV=\frac{165.65}{3540.00}\times 100 \]
\[ CV=4.68\% \]
Arrange the data in ascending order:
\(3310,\;3355,\;3450,\;3480,\;3480,\;3490,\;3520,\;3540,\;3550,\;3650,\;3730,\;3925\)
First quartile \(Q_1=P_{25}\):
\[ i=\frac{25}{100}\times 12 \]
\[ i=3 \]
Since \(i\) is an integer, average the 3rd and 4th values.
\[ Q_1=\frac{3450+3480}{2} \]
\[ Q_1=3465 \]
Second quartile \(Q_2=P_{50}\):
\[ i=\frac{50}{100}\times 12 \]
\[ i=6 \]
Since \(i\) is an integer, average the 6th and 7th values.
\[ Q_2=\frac{3490+3520}{2} \]
\[ Q_2=3505 \]
\(Q_2\) represents the median.
Third quartile \(Q_3=P_{75}\):
\[ i=\frac{75}{100}\times 12 \]
\[ i=9 \]
Since \(i\) is an integer, average the 9th and 10th values.
\[ Q_3=\frac{3550+3650}{2} \]
\[ Q_3=3600 \]
Five number summary:
\[ \text{Minimum}=3310 \]
\[ Q_1=3465 \]
\[ \text{Median}=3505 \]
\[ Q_3=3600 \]
\[ \text{Maximum}=3925 \]
Box plot idea: