The Mean of a Data Set
The mean of a set of numbers, sometimes simply called the average , is the sum of the data divided by the total number of data.
Example 1 :
Find the mean of the set { 2 , 5 , 5 , 6 , 8 , 8 , 9 , 11 } .
There are 8 numbers in the set. Add them all, and then divide by 8 .
2 + 5 + 5 + 6 + 8 + 8 + 9 + 11 8 = 54 8 = 6.75
So, the mean is 6.75 .
The Median of a Data Set
The median of a set of numbers is the middle number in the set [after the numbers have been arranged from least to greatest] -- or, if there are an even number of data, the median is the average of the middle two numbers.
Example 1 :
Find the median of the set { 2 , 5 , 8 , 11 , 16 , 21 , 30 } .
There are 7 numbers in the set, and they are arranged in ascending order. The middle number [the 4 th one in the list] is 11 . So, the median is 11 .
Example 2 :
Find the median of the set { 3 , 10 , 36 , 255 , 79 , 24 , 5 , 8 } .
First, arrange the numbers in ascending order.
{ 3 , 5 , 8 , 10 , 24 , 36 , 79 , 255 }
There are 8 numbers in the set -- an even number. So, find the average of the middle two numbers, 10 and 24 .
10 + 24 2 = 34 2 = 17
So, the median is 17 .
The Mode of a Data Set
The mode of a set of numbers is the number which occurs most often.
Example 1 :
Find the mode of the set { 2 , 3 , 5 , 5 , 7 , 9 , 9 , 9 , 10 , 12 } .
2 , 3 , 7 , 10 and 12 each occur once.
5 occurs twice and 9 occurs three times.
So, 9 is the mode.
Example 2 :
Find the mode of the set { 2 , 5 , 5 , 6 , 8 , 8 , 9 , 11 } .
In this case, there are two modes -- 5 and 8 both occur twice, whereas the other numbers only occur once.
Published on July 30, 2020 by Pritha Bhandari. Revised on June 9, 2022. Measures of central tendency help you find the middle, or the average, of a dataset. The 3 most common measures of central tendency are the mode, median, and mean. In addition to central tendency, the variability and distribution of your dataset is important to understand when performing descriptive statistics. A dataset is a distribution of n number of scores or values. In a normal distribution, data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The mean, mode and median are exactly the same in a normal distribution. A histogram of your data shows the frequency of responses for each possible number of books. From looking at the chart, you see that there is a normal distribution.Distributions and central tendency
Normal
distribution
The mean, median and mode are all equal; the central tendency of this dataset is 8.
Skewed distributions
In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other. The direction of this tail tells you the side of the skew
In a positively skewed distribution, there’s a cluster of lower scores and a spread out tail on the right. In a negatively skewed distribution, there’s a cluster of higher scores and a spread out tail on the left.
In this histogram, your distribution is skewed to the right, and the central tendency of your dataset is on the lower end of possible scores.
In a positively skewed distribution, mode < median < mean.
In this histogram, your distribution is skewed to the left, and the central tendency of your dataset is towards the higher end of possible scores.
In a negatively skewed distribution, mean < median < mode.
Mode
The mode is the most frequently occurring value in the dataset. It’s possible to have no mode, one mode, or more than one mode.
To find the mode, sort your dataset numerically or categorically and select the response that occurs most frequently.
Example: Finding the modeIn a survey, you ask 9 participants whether they identify as conservative, moderate, or liberal.To find the mode, sort your data by category and find which response was chosen most frequently.
To make it easier, you can create a frequency table to count up the values for each category.
| 2 |
| 3 |
| 4 |
Mode: Liberal
The mode is easily seen in a bar graph because it is the value with the highest bar.
When to use the mode
The mode is most applicable to data from a nominal level of measurement. Nominal data is classified into mutually exclusive categories, so the mode tells you the most popular category.
For continuous variables or ratio levels of measurement, the mode may not be a helpful measure of central tendency. That’s because there are many more possible values than there are in a nominal or ordinal level of measurement. It’s unlikely for a value to repeat in a ratio level of measurement.
Example: Ratio data with no modeYou collect data on reaction times in a computer task, and your dataset contains values that are all different from each other.| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 267 | 345 | 421 | 324 | 401 | 312 | 382 | 298 | 303 |
In this dataset, there is no mode, because each value occurs only once.
The median of a dataset is the value that’s exactly in the middle when it is ordered from low to high.
Example: Finding the medianYou measure the reaction times of 7 participants on a computer task and categorize them into 3 groups: slow, medium or fast.| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Medium | Slow | Fast | Fast | Medium | Fast | Slow |
To find the median, you first order all values from low to high. Then, you find the value in the middle of the ordered dataset—in this case, the value in the 4th position.
| Slow | Slow | Medium | Medium | Fast | Fast | Fast |
Median: Medium
In larger datasets, it’s easier to use simple formulas to figure out the position of the middle value in the distribution. You use different methods to find the median of a dataset depending on whether the total number of values is even or odd.
Median of an odd-numbered dataset
For an odd-numbered dataset, find the value that lies at the
| 287 | 298 | 345 | 365 | 380 |
The middle position is calculated using
That means the median is the 3rd value in your ordered dataset.
Median: 345 milliseconds
Median of an even-numbered dataset
For an even-numbered dataset, find the two values in the middle of the dataset: the values at the
| 287 | 298 | 345 | 357 | 365 | 380 |
The middle positions are calculated using
That means the middle values are the 3rd value, which is 345, and the 4th value, which is 357.
To get the median, take the mean of the 2 middle values by adding them together and dividing by 2.
Median: 351 milliseconds
Mean
The arithmetic mean of a dataset [which is different from the geometric mean] is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency because all values are used in the calculation.
Example: Finding the mean| 1 | 2 | 3 | 4 | 5 |
| 287 | 345 | 365 | 298 | 380 |
First you add up the sum of all values:
Then you calculate the mean using the formula
There are 5 values in the dataset, so n = 5.
Mean [x̄]: 335 milliseconds
Outlier effect on the mean
Outliers can significantly increase or decrease the mean when they are included in the calculation. Since all values are used to calculate the mean, it can be affected by extreme outliers. An outlier is a value that differs significantly from the others in a dataset.
Example: Mean with an outlierIn this dataset, we swap out one value with an extreme outlier.| 1 | 2 | 3 | 4 | 5 |
| 832 | 345 | 365 | 298 | 380 |
Due to the outlier, the mean [
Mean: 444 milliseconds
Population versus sample mean
A dataset contains values from a sample or a population. A population is the entire group that you are interested in researching, while a sample is only a subset of that population.
While data from a sample can help you make estimates about a population, only full population data can give you the complete picture.
In statistics, the notation of a sample mean and a population mean and their formulas are different. But the procedures for calculating the population and sample means are the same.
Sample mean formulaThe sample mean is written as M or x̄ [pronounced x-bar]. For calculating the mean of a sample, use this formula:
- x̄: sample mean
- : sum of all values in the sample dataset
- n: number of values in the sample dataset
- μ: population mean
- : sum of all values in the population dataset
- N: number of values in the population dataset
The 3 main measures of central tendency are best used in combination with each other because they have complementary strengths and limitations. But sometimes only 1 or 2 of them are applicable to your dataset, depending on the level of measurement of the variable.
- The mode can be used for any level of measurement, but it’s most meaningful for nominal and ordinal levels.
- The median can only be used on data that can be ordered – that is, from ordinal, interval and ratio levels of measurement.
- The mean can only be used on interval and ratio levels of measurement because it requires equal spacing between adjacent values or scores in the scale.
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To decide which measures of central tendency to use, you should also consider the distribution of your dataset.
For normally distributed data, all three measures of central tendency will give you the same answer so they can all be used.
In skewed distributions, the median is the best measure because it is unaffected by extreme outliers or non-symmetric distributions of scores. The mean and mode can vary in skewed distributions.
Frequently asked questions about central tendency
Sources in this article
We strongly encourage students to use sources in their work. You can cite our article [APA Style] or take a deep dive into the articles below.
This Scribbr article
Bhandari, P. [June 9, 2022]. Central Tendency | Understanding the Mean, Median & Mode. Scribbr. Retrieved October 21, 2022, from //www.scribbr.com/statistics/central-tendency/
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