4.1 Statistics II

4.1 Statistics II

4.1.1 Pie Charts
1.   A pie chart is a graphic representation of data using sectors of a circle. The size of each sector shows the percentage of each category of data it represents.

Example:

2.   The percentage of data represented by each sector can be calculated by using the formula below.
$\text{Percentage of data}=\frac{\text{angle of sector}}{{360}^{o}}×100%$

3.   To construct a pie chart, the angle of each sector can be calculated by using the formulae below.
$\text{Angle of sector}=\frac{\text{data of each category}}{\text{overall data}}×{360}^{o}$

4.   Data can also be represented by pictograms, bar charts and line graphs.

5.   The choice of any these representations depends very much on the suitability of the data and the aim we wish to achieve in representing the data.

4.1.2 Mode, Median and Mean
1.   The mode of a set of data is the value of item which occurs most frequently.
Example:
3, 7, 6, 9, 7, 1, 5, 7, 2
Mode = 7

2.   When a set of data is given in a frequency table, the value or item which has the highest frequency is the mode.

3.   The median of a set of data is the value located in the middle of the set when the data is arranged in numerical order.
- If the total number of data is odd, then the median is the value in the middle of the set.
- If the total number of data is even, then the median is the average of the two middle values of the set

Example 1:
Find the medians of the following sets of data:
(a) 10, 9, 11, 6, 5, 8, 7
(b)   10, 9, 11, 6, 5, 8, 7,12

Solution:
(a) Number of data values = 7 ← (Odd number)
Rearranging the data in order of magnitude:
5, 6, 7, 8, 9, 10, 11
Therefore, median = 8
(b)   Number of data values = 8 ← (Even number)
Rearranging the data in order of magnitude:
5, 6, 7, 8, 9, 10, 11, 12

$\begin{array}{l}\therefore \text{Median}=\frac{8+9}{2}\\ \text{}=8.5\end{array}$

4.   When a set of data is given in a frequency table, the value situated in the middle position of the data is the median.

5.   The mean of a set of data is obtained by using formula below.

$\text{Mean =}\frac{\text{sum of all values of data}}{\text{total number of data}}$

Example:
Find the mean of the following sets of data items:
-5, -2, -1, 7, 4, 9
Solution:
$\begin{array}{l}\text{Mean}=\frac{\left(-5\right)+\left(-2\right)+\left(-1\right)+7+4+9}{6}\\ \text{}=\frac{12}{6}\\ \text{}=2\end{array}$

6.   When data is given in a frequency table, the mean can be found by using the formula below.

$\text{Mean =}\frac{\text{sum of}\left(\text{value}×\text{frequency}\right)}{\text{total frequency}}$

Example:
The table below shows the scores obtained by a group of players in a game.
 Score 1 2 3 4 5 Frequency 5 12 8 15 10
Find the mean of the scores.

Solution:
$\begin{array}{l}\text{Mean}\\ =\text{}\frac{\text{sum of}\left(\text{score}×\text{frequency}\right)}{\text{total frequency}}\\ =\frac{\left(1×5\right)+\left(2×12\right)+\left(3×8\right)+\left(4×15\right)+\left(5×10\right)}{5+12+8+15+10}\\ =\frac{163}{50}\\ =3.26\end{array}$