The mean平均値 is the representative value found by dividing the sum of all values by the number of values.
A representative value sums up the center or typical shape of a data set with just one or a few numbers. Because the mean uses every value, a single value far from the rest (an outlier) pulls it strongly in that direction.
AnalogyPicture leveling block towers of different heights until they all stand equally tall; that shared height is the mean.
Exam points
ExamWhen finding the mean 平均値, add a repeated value into the total once for every time it appears, then divide by the count including the repeats.
The median中央値 is the value that sits in the middle when the data are arranged in order of size.
Sort from smallest to largest
Check whether the count is odd or even
Odd: the middle one; even: the average of the middle two
If the count is odd, the single middle value is the median; if even, the median is the average of the two middle values.
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With an odd number of values, the median is the single middle value; with an even number, it is the average of the two middle values. Since it depends only on position in the sorted order, making the largest value even larger leaves the median unchanged.
AnalogyIn a line of people sorted by height, the height of the person standing exactly in the middle is the median.
Exam points
ExamIn data containing an outlier 外れ値, the mean 平均値 shifts strongly toward the outlier, while the median 中央値 stays at the middle of the sorted order, so the two can end up far apart.
In Japanese, the median also appears as the loanword メジアン, another name for 中央値 that refers to exactly the same value.
Statistical materials in Japan use the native term 中央値 and the loanword メジアン interchangeably. The calculation is also the same as for 中央値: sort the values by size and take the one in the middle.
How to rememberThe English word median means "in the middle," and the strip dividing a highway is also called the median.
The mode最頻値 is the value that appears most often in the data.
Representative value
How to find it
For data 2, 2, 3, 5, 8
Mean
Divide the total by the count
20 ÷ 5 = 4
Median
Middle value after sorting
3rd value = 3
Mode
Most frequent value
2 (appears twice)
Count how many times each value appears and pick the value with the highest count. No arithmetic is involved, so the mode also works for non-numeric data such as favorite colors. If two or more values tie for the highest count, there are two or more modes.
AnalogyThe item a store sold the most of on a given day is the mode of that day's sales record.
How to rememberThe kanji 頻 in 最頻値 is the one used in words for "frequent" and "frequency," so it signals how often something occurs.
Exam points
Mix-upThe mode 最頻値 is decided by counting occurrences, unlike the mean 平均値, which adds values together, and the median 中央値, which depends on position in the sorted order.
In Japanese, the mode also appears as the loanword モード, another name for 最頻値 meaning the value that appears most often.
It is found the same way as 最頻値: count how often each value appears and choose the most frequent one. Whether a document writes 最頻値 or モード, it refers to the same value.
How to rememberLink it to mode in fashion: what is in style is what the most people are choosing.
Variance分散 is the average of the squared differences between each value and the mean, and it shows how widely the data are spread.
Find the mean
Subtract the mean from each value (deviation)
Square each deviation
Average the squares = variance
The average of the squares is the variance.
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A measure of dispersion is a number that shows how far the data scatter around their center. The difference between each value and the mean is its deviation, and because deviations simply added together cancel out to 0, they are squared before being averaged. If every value is the same, all deviations are 0, so the variance is 0 too.
AnalogyIt resembles scoring a round of darts by how far each dart lands from the center of the whole cluster of darts, with darts far from that center counting much more heavily.
Exam points
ExamIn a frequency distribution 度数分布 graph, the more the frequencies cluster near the mean and the fewer scores lie far from it, the smaller the variance 分散.
The standard deviation標準偏差 is the positive square root of the variance, expressing spread in the same unit as the original data.
Data
Mean
Variance
Standard deviation
4, 4, 6, 6
5
1
1
2, 2, 8, 8
5
9
3
Because variance squares the deviations, its unit is squared as well (for test scores, "points squared"). Taking the square root brings the standard deviation back to the original unit, so it can be set side by side with differences from the mean.
AnalogyJust as you recover a side length from an area, the standard deviation brings the squared-up variance back to the original ruler.
How to rememberRead it as the "standard" deviation: a rough gauge of how far a typical value sits from the mean.
Exam points
ExamDividing a value's difference from the mean 平均値 by the standard deviation 標準偏差 shows how many standard deviations it lies from the mean, so positions can be compared even between groups with different spreads.
A deviation value偏差値 is a number that shows position within a group on a scale where the mean is 50 and each standard deviation is worth 10.
Find the difference from the mean
Divide the difference by the standard deviation
Multiply by 10 and add 50
1.5 × 10 + 50 gives a deviation value of 65.
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As a formula, deviation value = 50 + 10 × (value − mean) ÷ standard deviation. A value exactly at the mean scores 50, one standard deviation above scores 60, and one below scores 40. Tests with different maximum scores or difficulty can thus be compared on the same scale.
AnalogyLike exchanging different national currencies into one currency before comparing, it converts each group's scores onto a common scale before comparing them.
Exam points
ExamWhether a deviation value 偏差値 is high or low depends on the difference from the mean 平均値 divided by the standard deviation 標準偏差. The same score can give a different deviation value when the group's mean or standard deviation differs.
This unit covers three representative values (mean, median, and mode); the loanwords メジアン and モード are just the English names for 中央値 and 最頻値.
A deviation is each value minus the mean; averaging the squared deviations gives the variance, and its positive square root gives the standard deviation.
The more tightly values gather around the mean, the smaller both the variance and the standard deviation; if all values are equal, both are 0.
A deviation value is calculated as 50 + 10 × (value − mean) ÷ standard deviation, and a value exactly at the mean gets 50.