Key takeaway: Mean, median and mode are three ways to name the center of a data set: the evenly shared total, the middle value of the sorted list, and the value that appears most often.
Grade 66.SP.A.3Statistics and Data
Definition of Mean, Median and Mode
The mean, median and mode each try to capture a typical value of a data set, but they do it in different ways. The mean shares the total out equally, the median finds the middle position, and the mode picks the most popular value.
To get the mean, add everything up and split the sum by the number of values. To get the median, put the values in order and take the one in the middle; with an even count, average the two middle ones. The mode is the value that shows up most; a set can have one, several, or none. Which center fits best depends on the data, since a single huge value drags the mean upward yet leaves the median almost where it was.
Formula and notation
For \(n\) values \(x_1, x_2, \text{…}, x_n\), the mean is written with a bar:
\[\,\overline{x} = \frac{x_1 + x_2 + \text{…} + x_n}{n}\]
The median is the middle entry of the sorted list, and the mode is the entry that is repeated most.
Key facts
- The mean acts like a balance point, so every value pulls on it.
- The median depends only on position, so extreme values hardly move it.
- A data set can have no mode, one mode, or several modes.
- The mean, median and mode often disagree, so quote the one that fits the data.
Mean, Median and Mode example
A baseball team scored 5, 9, 3, 12, 9, 6, 14, 5, 9 and 8 runs in ten games. Find the mean, the median and the mode of the scores.
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Add the scores: \(5 + 9 + 3 + 12 + 9 + 6 + 14 + 5 + 9 + 8 = 80\). Divide by 10 games to get 8.
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Order them: 3, 5, 5, 6, 8, 9, 9, 9, 12, 14. The two in the middle are 8 and 9, so the middle value is \(\frac{8 + 9}{2} = 8.5\).
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The score 9 occurs three times , more often than any other.
The mean is 8, the median is 8.5 and the mode is 9 runs.
Common mistake
Watch out
Without sorting, Tomas takes the fifth and sixth numbers of the list as written, 9 and 6, and answers that the median is 7.5.
Why it is wrong
The median depends on order, so the list must be sorted first. In the sorted list the fifth and sixth numbers are 8 and 9, which gives 8.5. Skipping the sort picks values from the wrong positions.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Statistics and Data