The average (or mean) is a measure of central tendency of quantitative data and is widely used as a representative value for the data. The mean is the sum of all data and divided by the number of data, which implies the center of gravity of the data. The mean is expressed as \(\mu\) (read mu), and the mean of (Data 4.1) is obtained as follows: $$ \text{Mean} = \mu = \frac{63 + 60 + 65 + 55 + 77}{5} = \frac{320}{5} = 64 $$ When \(n\) number of data is expressed as \(x_1 , x_2 , ... , x_n \), mean is expressed by the following formula.

$$ \small \mu = \frac{1}{n} \sum_{i=1}^n x_i $$

In general, the mean is very appropriate as a representative value of the data, but when there is a very large or a small value in the data, it is greatly affected by this extreme value. In this case, a median can be used. The median is the value in the middle when the data are sorted in order. In (Data 4.1), the number of data is 5 which is an odd number, and the 3rd (\(\frac{n+1}{2}\)) number when data is sorted in ascending order is the median as follows:

If the number of data is 6 which is an even number, how do we find the median? In this case, the median of the data is calculated as the average of the 3rd (=\(\frac{n}{2}\) ) and 4th (=\(\frac{n+2}{2}\) ) of the sorted data.

Generally, a median is expressed as \(m\), and, if the number of data is \(n\) , it is calculated as follows:

1) Data are sorted in ascending order.
2) Check whether the number of data is an odd number or an even number.
3) If \(n\) is odd, \(m\) = \( (\frac{n+1}{2})^{th}\) data in sorted data.
\(\quad \;\)If \(n\) is even, \(m\) = Average of (\(\frac{n}{2})^{th}\) data \((\frac{n+2}{2})^{th}\) data in sorted data.

In order to see the overall distribution of the weight data, a stem and leaf plot or a histogram discussed in Chapter 3 can be considered, but a dot graph is more useful. In a dot graph, after obtaining the minimum and maximum values of data, the position of each data is calculated on the horizontal axis, and displayed as a dot.

<Figure 4.1> is a dot graph for (Data 4.1). In proportion to the minimum value of 55 and the maximum value of 76, each data is displayed by a dot. The green line is the mean and the red line is the median. In this data, the mean is located slightly to the right of the median because 77 of the data is located to the right of the other four data. That is, the mean is more sensitive to an extreme than the median.

If there are lots of data, it is time-consuming and difficult to obtain the mean and median manually as above. Let's find the representative value of the data using 『eStatM』 software.

When a frequency table is given rather than the raw data, the mean can be obtained approximately as follows using the middle values of each class interval.

First, find the middle value of each class interval. Then, it is assumed that each class has the middle value as many as the frequency, and the mean is obtained using this approximated data.

Mean is calculated as follows: $$ \small \begin{align} \text{Mean} &= \frac{65+65+75+75+75+75+75+85+85+85+85+85+85+85+85+85+85+95+95+95}{20} \\ &= \frac{65 \times 2 + 75 \times 5 + 85 \times 10 + 95 \times 3} {20} \\ &= \frac{1640}{20} = 82 \end{align} $$

The degree to which data are scattered is called a dispersion. A simple measure of the dispersion is a range which is the maximum minus the minimum. $$ \text{Range} = \text{Maximum - Minimum} $$ In (Data 4.2), the maximum value is 10 and the minimum value is 4, so the range is 22. $$ \text{Range} = \text{77 - 55 = 22} $$

Since the range is too sensitive to extreme values, a variance or a standard deviation is generally used to measure the dispersion. The variance is obtained by squaring the distance between each data value and the mean, and dividing it by the number of data. Therefore, when the data are scattered far from the mean, the variance is large, and when the data are clustered around the mean, the variance is small. The variance is expressed as \(\sigma^2\) (read as sigma squared).

The mean of the data in (Data 4.2) is as follows: $$ \text{Mean} \quad \mu ~=~ \frac{6+8+7++4+10}{5} ~=~ \frac{35}{5} ~=~ 7 $$

The variance is calculated by squaring the distances from the mean to each data value to find the sum, and then finding the mean. That is, it is the average of squared distances. $$ \begin{align} \text{Variance} \quad \sigma^{2} &~=~ \frac{ (6-7)^2 + (8-7)^2 + (7-7)^2 + (4-7)^2 + (10-7)^2} {5} \\ &~=~ \frac{20}{5} ~=~ 4 \end{align} $$ When \(n\) number of data is expressed as \(x_1 , x_2 , ... , x_n\) and the mean is expressed as \(\mu\), the variance can be expressed by the following formula. $$ \begin{align} \text{Variance} \quad \sigma^{2} ~=~ { {1 \over n} {\sum _{i=1} ^{n} (x_{i} - \mu )^{2}} } ~~~~ (n:~자료수) \\ \end{align} $$

The standard deviation is defined as the square root of the variance and denoted by \(\sigma\). The variance is not easy to interpret practically because it is the average of the squared distances, but the standard deviation is the square root of the variance, so it can be interpreted as a measure of the average distance between each data value and the mean. $$ \text{Standard deviation} \quad \sigma ~=~ \sqrt{\sigma^2} \\ $$ The standard deviation of (Data 4.2) is \(\sigma\) = \(\sqrt{\sigma^2}\) = \(\sqrt{4}\) = 2 이다.

In the previous section, when a frequency table was given rather than the raw data, the mean was approximated using the middle value of each class interval. The standard deviation is calculated in a similar way.

First, find the middle value of each class. Then, it is assumed that each class has the middle value as many as the frequency, and the average is obtained using this approximate data.

Mean is calculated as follows: $$ \text{Mean} ~=~ \frac{65 \times 2 + 75 \times 5 + 85 \times 10 + 95 \times 3} {20} ~=~ \frac{1640}{20} ~=~ 82 $$

The variance and standard deviation are calculated in a similar way. $$ \small \begin{align} &\text{Variance} \qquad \qquad \sigma^2 ~=~ \frac{(65-82)^2 \times 2 + (75-82)^2 \times 5 + (85-82)^2 \times 10 + (95-82)^2 \times 3} {20} \\ &\qquad\qquad\qquad\qquad \;\; ~=~ \frac{1420}{20} ~=~ 71 \\ &\text{Standard deviation} \qquad \sigma ~=~ \sqrt{\sigma^2} ~=~ \sqrt{71} ~=~ 8.43 \end{align} $$

Using ‘Frequency Distribution Polygon - Relative Frequency Comparison’ of 『eStatM』, the approximate mean and standard deviation of the frequency table can be obtained as shown in <Figure 4.6>. After entering the left value of the class interval and ‘Frequency 1’, click the [Execute] button.

Just as the variance is used as a measure of dispersion in one quantitative variable, the following covariance is used in two quantitative variables. When \(n\) number of x and y data are expressed as \( (x_1 , y_1 ), (x_2 , y_2 ), ... , (x_n , y_n ) \), and the mean is expressed as \( (\mu_x , \mu_y )\), the covariance \(\sigma_{xy}\) can be expressed by the following formula. $$ \text{Covariance} \quad \sigma_{xy} ~ =~ \frac{1}{n} \sum _{i=1} ^{n} (x_{i} - \mu_x ) (y_{i} - u_y ) \qquad (n:\text{ number of data} ) $$

Covariance implies the total average of the values obtained by multiplying the x-axis distance and the y-axis distance between each data point and the mean point of data. Therefore, if there are many points on the upper right and lower left of the mean point, the covariance has a positive value, indicating a positive correlation. If there are many points on the upper left and lower right of the mean point, the covariance has a negative value, indicating a negative correlation. However, since covariance can increase in value depending on the unit of data, the following correlation coefficient denoted as \(\rho\) is used as a measure of correlation. $$ \text{Correlation coefficient} \quad \rho ~ =~ \frac{\sigma_{xy}}{\sigma_x \sigma_y} $$

The correlation coefficient is a variation of the covariance and can only have values between -1 and +1. When the correlation coefficient is close to +1, the two variables are said to have a strong positive correlation, and when the correlation coefficient is close to -1, it is said to have a strong negative correlation. When the correlation coefficient is close to 0, there is no correlation between the two variables.

If the correlation is strong, a straight line that can explain the relationship between the variables is obtained, which is called a regression line. A detailed explanation of the regression line is covered in university level of statistics.