A small number of values used to summarize the characteristics of an entire data set are called statistics. In this lesson, you will learn representative values that describe the center, variance and standard deviation that describe spread, quartiles and box-and-whisker plots that summarize the shape of a distribution, and the normal distribution that appears widely in nature and society.
Representative Values — Mean・Median・Mode
・Mean: The total divided by the number of data values. It uses all data but is sensitive to outliers.
・Median: The middle value when the data is arranged from smallest to largest. It is resistant to outliers. When there is an even number of data values, it is the mean of the two middle values.
・Mode: The value that appears most often. It is the only representative value that can also be used for qualitative data.
Measuring Spread — Variance and Standard Deviation
Even when two data sets have the same mean, it makes a great difference whether the data is concentrated near the mean or widely spread out. The measures of spread are variance and standard deviation.
① The value obtained by subtracting the mean from each data value is called the deviation. The sum of the deviations is always 0.
② The mean of the squared deviations is the variance.
③ The positive square root of the variance is the standard deviation. It returns to the same unit as the original data, making it easier to interpret.
・Mean ＝ (4＋8＋10＋12＋16)÷5 ＝ 50÷5 ＝ 10 points
・Deviations ＝ −6、−2、0、2、6 (confirm that their sum is 0)
・Squared deviations ＝ 36、4、0、4、36 → total 80
・Variance ＝ 80÷5 ＝ 16
・Standard deviation ＝ √16 ＝ 4 points
Saying “the mean is 10 points and the standard deviation is 4 points” conveys that the scores are spread around 10 points, generally within a range of about ±4 points.
Quartiles and Box-and-Whisker Plots
The values at the positions that divide data arranged from smallest to largest into four equal parts are called quartiles. From the smallest, they are the first quartile (Q1), second quartile (Q2＝median), and third quartile (Q3). Q3−Q1 is called the interquartile range (IQR) and represents the spread of the middle 50% of the data.
A box-and-whisker plot displays the five values—the minimum, Q1, median, Q3, and maximum—known as the five-number summary, in one diagram. The box represents Q1 to Q3, the line inside the box represents the median, and the whiskers represent the minimum and maximum. It is useful for comparing the distributions of several groups side by side.
・Median (Q2) ＝ the 4th value, 60 points
・Q1 ＝ median of the lower group {40, 50, 55} ＝ 50 points
・Q3 ＝ median of the upper group {70, 80, 90} ＝ 80 points
・Interquartile range ＝ 80−50 ＝ 30 points
When drawn as a box-and-whisker plot, the box ranges from 50 to 80 points, the line inside the box is at 60 points, and the whiskers extend to 40 and 90 points.
Normal Distribution — The Most Common Distribution in the World
Many data sets, such as height and measurement errors, follow a symmetric, bell-shaped distribution in which values are most common near the mean and become less common as they move away from it. This is called a normal distribution. A normal distribution has useful properties: about 68% of all data lies within the range of the mean ±1 standard deviation, and about 95% lies within the range of the mean ±2 standard deviations. For example, in a test with a mean of 60 points and a standard deviation of 10 points, about 68% of people are estimated to score between 50 and 70 points, and about 95% between 40 and 80 points. The “deviation score” used in practice examinations is also an indicator based on a normal distribution, with the mean converted to 50 and the standard deviation to 10.
- Deviation＝data−mean. The sum of deviations is always 0. Therefore, square them and take their mean (＝variance).
- Standard deviation＝√variance. Its unit is the same as that of the original data, making it easy to interpret.
- Quartiles divide data into four equal parts. Q3−Q1＝interquartile range (spread of the middle 50%).
- A box-and-whisker plot shows the five-number summary (minimum・Q1・median・Q3・maximum). It is effective for comparing distributions.
- In a normal distribution, about 68% lies within ±1σ of the mean and about 95% within ±2σ.
Practice Problems
- Find the mean, variance, and standard deviation of the data 6、8、10、12、14 (you may use √2≒1.41).
- Find the first quartile, median, and third quartile of the scores of nine students: 10、20、30、40、50、60、70、80、90.
- In a population following a normal distribution with a mean of 160cm and a standard deviation of 5cm, approximately what percentage of people are between 155cm and 165cm tall?
Answers・Explanations
- Answer: Mean 10, variance 8, standard deviation 2√2≒2.83
Explanation: Mean＝50÷5＝10. The deviations are −4、−2、0、2、4, and the sum of their squares is 16＋4＋0＋4＋16＝40. Variance＝40÷5＝8, and standard deviation＝√8＝2√2≒2.83. - Answer: Q1＝25, median＝50, Q3＝75
Explanation: The median is the 5th value, 50. The median of the lower group {10,20,30,40} is (20＋30)÷2＝25, and the median of the upper group {60,70,80,90} is (70＋80)÷2＝75. The interquartile range is 75−25＝50. - Answer: About 68%
Explanation: 155〜165cm is the range “mean ±1 standard deviation” (160±5). In a normal distribution, about 68% lies within this range. Within ±2σ (150〜170cm), about 95% lies in the range.