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3.2 Standard deviation

Standard deviation determines the variations by all values in a group into consideration. The size of the variations is determined by how far the arithmetic average is away from individual values in a group.
The arithmetic average of group D (1, 1, 4, 7, 7）is 4. When one subtracts the arithmetic average from each value, then the solution is (1-4 = -3), (1-4 = -3), (4-4=0), (7-4=3) and (7-4=3). When one averages the deviation of values from the arithmetic average, the rough standard of variations can be obtained. But when one adds -3, -3, 0, 3, 3, then the solution is 0. Therefore, it becomes an ingenuity again.
For example, there is a method to take the absolute value of a number or square of a number and to then delete the minus sign. When squaring, the solution is 9, 9, 0, 9, 9, and the average is 7.2 by dividing by 5. However, when the original unit is cm, then it is cm2 by squaring, therefore, one extracts 7.2 and restores it, then the size of variation comes to √7.2 cm2≒2.68 cm.

(1) Formula of standard deviation
σ＝√Σ (Xi−X)2／n

Next, by looking at group E (1, 4, 4, 4, 7), the arithmetic average is 4. When one subtracts the arithmetic average from each value, the values are (1-4 = -3), (4-4 = 0), (4-4 = 0), (4-4 = 0) and (7-4=3). When one averages the size of deviation from the arithmetic average, the rough standard of variations can be obtained. But when one adds -3, 0, 0, 0, 3, then the solution is 0. Therefore, we square each value and average the values to get an average of 3.6 (by dividing by 5).
However, when the original unit is cm, it becomes cm2 by squaring, therefore we extract 3.6 and restore it, then the size of variations is √3.6 cm2≒1.89 cm. Therefore, group D is bigger than group E in terms of variations.
Below, I will examine the characteristics seen from the relational database of “Jingoro Sahashi” by using the standard deviation formula.

花村嘉英（2017）「日本語教育のためのプログラム」より英訳 translated by Yoshihisa Hanamura