How to Find the Measures of Central Tendency? (+FREE Worksheet!)

Median: The middle value that separates the higher half from the lower half. Numbers are arranged in either ascending or descending order. The middle number is then taken.

Mode: the most frequent value. It is used to show the most popular option and is the highest bar in the histogram.

To decide which one to use, the following should be considered:

• The mean gives all values the same importance, even very large or very small values, while the median focuses more on the values that are in the middle of the data set; So, the mean can use the data more fully. However, as mentioned, the mean can be strongly influenced by one or two very large or very small values.
• There is always only one value for the median or mean on a data set, but a set can have more than one mode.
• Among the Measures of Central Tendency, the mode is less used than the mean and median. However, in some cases, the mode can be significantly useful.

Finding the Measures of Central Tendency: Example 1:

Find the mean, median, and mode of the data set. \(10, 11, 25, 16, 16, 46, 29, 35\)

Solution: Mean: first add the values \(10+11+25+16+16+46+29+35=188\)

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Divide by \(8\), the number of values. Mean\(=\frac{188}{8}=23.5\)

Median: Order the data from least to greatest. \(10, 11, 16, 16, 25, 29, 35, 45\)

Average the two middle values. \(=\frac{16+25}{2}=20.5\)

 So, \(20.5\) is median.

Mode: the value \(16\) occurs two times. So, \(16\) is the mode.

Finding the Measures of Central Tendency Example 2:

Find the mean, median, and mode of the data set. \(22, 35, 45, 53, 52, 35, 11, 16, 35, 11\)

Solution: Mean: first add the values \(22+35+45+53+52+35+11+16+35+11=315\)

Divide by \(10\), the number of values. Mean\(=\frac{315}{10}=31.5\)

Median: Order the data from least to greatest. \(11, 11, 16, 22, 35, 35, 35, 45, 52, 53\)

Average the two middle values. \(=\frac{35+35}{2}=35\)

 So, \(35\) is median.

Mode: the value \(35\) occurs three times. So, \(35\) is the mode.

Finding the Measures of Central Tendency Example 3:

Find the mean, median, and mode of the data set. \(9, 2, 5, 21, 16, 5, 36, 13, 10\)

Solution: Mean: first add the values \(9+2+5+21+16+5+36+13+10=117\)

Divide by \(9\), the number of values. Mean\(=\frac{117}{9}=13\)

Median: Order the data from least to greatest. \(2, 5, 5, 9, 10, 13, 16, 21, 36\)

The middle value is the median. So, median\(=10\)

Mode: the value \(5\) occurs two times. So, \(5\) is the mode.

Finding the Measures of Central Tendency Example 4:

Find the mean, median, and mode of the data set. \(25, 36, 39, 8, 17, 45, 60, 1, 36, 42, 10\)

Solution: Mean: first add the values \(25+36+39+8+17+45+60+1+36+42+10=319\)

Divide by \(11\), the number of values. Mean\(=\frac{319}{11}=29\)

Median: Order the data from least to greatest. \(1, 8, 10, 17,25, 36, 36, 39, 42, 45, 60 \)

The middle value is the median. So, median\(=36\)

Mode: the value \(36\) occurs two times. So, \(36\) is the mode.

Exercises for Finding the Measures of Central Tendency

Calculate the mean, median, and mode for the following data sets.

1. Points scored by a basketball player: {\(8, 3, 17, 26, 13, 3, 30\)}
2. Marks on a set of tests: {\(64, 88, 95, 75, 69, 88, 70, 77\)}
3. Waiting time, in minutes: {\(13, 19, 11, 19, 7, 32, 45, 33\)}
4. Monthly rent \($\): {\(630, 585, 670, 710, 670, 600, 590\)}

1. Mean: \(14.29\) Median: \(13\) Mode: \(3\)
2. Mean: \(78.25\) Median: \(76\) Mode: \(88\)
3. Mean: \(22.38\) Median: \(19\) Mode: \(19\)
4. Mean: \(636.43\) Median: \(630\) Mode: \(670\)

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For a complete grade-level workbook that covers data and statistics alongside the rest of pre-algebra, Pre-Algebra for Beginners walks you through statistics topics with clear examples and try-it-yourself problems. For more practice with data and probability at middle-school level, Mastering Grade 6 Math includes plenty of worked exercises.

Frequently Asked Questions

What are the three measures of central tendency?

Mean, median, and mode. The mean is the arithmetic average – add all values and divide by how many there are. The median is the middle value once the data is sorted in order. The mode is the value that appears most often. Each one summarizes a different sense of “typical” – the mean balances the whole set, the median splits it in half, and the mode picks the most common point.

How do you calculate the mean?

Add up all your values, then divide by the number of values. For the test scores \(72, 85, 91, 88, 73\), the sum is \(72 + 85 + 91 + 88 + 73 = 409\). There are 5 values, so the mean is \(409 / 5 = 81.8\). The formula is \[\bar{x} = \frac{\sum x_i}{n}.\]

How do you find the median?

Sort the data from smallest to largest, then find the middle. For an odd count of values, the median is the single middle value. For an even count, the median is the average of the two middle values. Example: sorted scores \(72, 73, 85, 88, 91\) – the middle (3rd) value is \(85\). For \(72, 73, 85, 88\), the median is \((73 + 85)/2 = 79\).

What’s the mode of a data set?

The mode is the value that appears most often. The set \(\{2, 3, 3, 5, 7\}\) has mode \(3\). A set can have no mode (every value appears once), one mode (unimodal), or two or more modes (bimodal, multimodal). For \(\{4, 4, 7, 7, 9\}\), both \(4\) and \(7\) are modes.

When is the mean misleading?

When your data has extreme values (outliers). Consider salaries of \(40{,}000, 42{,}000, 45{,}000, 47{,}000, 50{,}000, 2{,}000{,}000\) – the mean is about \(370{,}667\), but five of six people earn under \(50{,}000\). The single huge salary drags the mean far above what’s typical. The median (\(46{,}000\)) gives a much more honest sense of the center.

Why is the median better for skewed data?

The median only depends on the middle position, not on how extreme the largest or smallest values are. Swapping the biggest salary in the example above from \(2{,}000{,}000\) to \(20{,}000{,}000\) doesn’t change the median at all. That makes the median the right measure to report for income, home prices, and other right-skewed distributions.

Can the mean, median, and mode all be different?

Yes – and they usually are. For the set \(\{1, 2, 2, 3, 10\}\), the mean is \(3.6\), the median is \(2\), and the mode is \(2\). For symmetric distributions (like a perfect bell curve), all three are equal. The bigger the gap between them, the more skewed your data.

What’s the range, and how is it different from these measures?

The range is the largest value minus the smallest. It measures spread, not center. For \(72, 85, 91, 88, 73\), the range is \(91 – 72 = 19\). Range and central tendency together give a fuller picture: the mean tells you the typical value, the range tells you how spread out the data is.

Which measure should I report?

It depends on the data. Use the mean for roughly symmetric numerical data without big outliers (heights, test scores in a normal class). Use the median for skewed data or when outliers are present (incomes, house prices, reaction times). Use the mode for categorical data (favorite color, most common shoe size) where averaging doesn’t make sense.

Where do these measures show up on tests?

Middle-school state tests from grade 6 up, the SAT, ACT, GED, HiSET, GRE, ASVAB, and most college placement tests. Common question types: compute mean/median/mode from a small data set, compare two data sets, find a missing value given the mean, or predict how an outlier affects each measure.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• How to find mean, median, mode, and range
• How to find mean absolute deviation
• How to find range, quartile, and IQR
• How to interpret histograms
• How to graph a box plot