# Mean, Median, and Mode: Measures of central tendency (2023)

## What is the central tendency?

Central trend measurements are summaryThe statisticsrepresenting the midpoint or typical value of a data set. Examples of these measures are mean, median, andModus. These statistics indicate where most of the values ​​fall in a distribution, and are also known as the central position of a distribution. You can think of central tendency as the tendency of data points to cluster around a mean value.

In statistics, the three most common measures of central tendency are mean, median, and mode. Each calculates the center using a different method. Choosing the best measure of central tendency depends on the nature of your data. In this post, I explore mean, median, and mode as measures of central tendency, show you how to calculate them, and how to determine which one works best for your data.

## Localization of the actions of the central trend

Most articles on mean, median, and mode focus on how to calculate these measures of central tendency. I will certainly do that, but I will start with a slightly different approach. My philosophy in my blog is to help you grasp statistics intuitively by focusing on concepts. So I'll start by graphing the central point of several datasets - so you understand the goal. Then we proceed to select the best measure of central tendency for your data and calculations.

The following three distributions represent different data conditions. In each distribution, find the region that the most common values ​​fall in. Even if the forms and types of data are different, you can find this central trend. This is the region in the distribution where the most common values ​​are found. These examples cover mean, median, and mode.

When the charts are highlighted, you can see where most of the values ​​occur. That's the concept. Measures of central tendency represent that idea with value. Next, you'll learn that as the distribution and type of data change, so does the best measure of central tendency. Consequently, you must know the nature of your data and graph it before choosing between mean, median and mode!

Related posts:Guide to data types and how to represent them graphically

Whether you use mean, median, or mode, central tendency is just one characteristic of a distribution. Another aspect is the variability around this central value. While measures of variability are the subject of another article (link below), this property describes how far the data points tend to fall from the center. The graph below shows how distributions with the same central tendency (mean = 100) can actually be very different. The panel on the left shows a distribution that is tightly clustered around the mean, while the distribution on the right is more spread out. It is important to understand that the central tendency summarizes only one aspect of a distribution and that, on its own, it provides an incomplete picture.

Related post:Variability measures: range, interquartile range, variance, and standard deviation

(Video) Mean, Median, and Mode: Measures of Central Tendency: Crash Course Statistics #3

## Mean

The mean is the arithmetic mean, and it's probably the measure of central tendency that you're most familiar with. Calculating the mean is very easy. You simply add up all the values ​​and divide by the number of observations in your data set.

The calculation of the mean includes all values ​​in the data. If you change a value, the mean changes. However, the mean does not always precisely locate the center of the data. Notice the histograms below where I show the mean in the distributions.

With a symmetrical distribution, the mean precisely locates the midpoint.

With a skewed distribution, however, the mean can miss the target. In the histogram above, it's starting to fall out of the central area. This problem occurs becauseRunawayhave a significant impact on the mean as a measure of central tendency. Extreme values ​​in an extended edge pull the mean away from the center. As the distribution becomes more skewed, the mean gets further from the center. Therefore, it's best to use the mean as a measure of central tendency when you have a symmetric distribution. More on this topic when we look at the mean vs median!

In statistics, we generally use the arithmetic mean, which is the type I'm going to discuss in this post. However, there are other types of averages, such as B. the geometric mean. Read my post about thelearn geometric mean when it's a better measure. Use aweighted meanif you need to attach different meanings to the values.

When to use the mean: Symmetrical distribution,Continuous data

Related posts:Using histograms to understand your dataandWhat is the mean?

## Median

The median is the mean. It is the value that splits the data set in half, making it a natural measure of central tendency.

To find the median, order your data from smallest to largest, then find the data point that has an equal number of values ​​above and below. The method of finding the median varies slightly depending on whether your dataset has an even or odd number of values. I show you how to find the median for both cases. In the following examples, I use integers for simplicity, but you can use decimals.

In the dataset with the odd number of observations, notice that the number 12 has six values ​​above and six below. Therefore, 12 is the median of this dataset.

(Video) Mean, Median, Mode, and Outliers: Measures of Central Tendency

If there is an even number of values, count up to the two innermost values ​​and then take the mean value. The average of 27 and 29 is 28. Therefore, 28 is the median of this dataset.

Runawayand skewed data will have a smaller oneEffecton the mean vs. median as a measure of central tendency. To understand why, imagine we have the median dataset below and find that the median is 46. However, we discover data entry errors and need to change four values ​​that are shaded in the Median Fixed dataset. We're going to make them all significantly higher, so we now have a skewed distribution with large outliers.

As you can see, the median doesn't change at all. It's still 46. When comparing the mean to the median, the mean depends on all values ​​in the data set, while the median does not. Consequently, when some of the values ​​are more extreme, the effect on the median is smaller. Of course, with other types of changes, the median may change. If the distribution is skewed, the median is a better measure of central tendency than the mean.

Related post:skewed distributions

### Mean vs. median as a measure of central tendency

Now let's compare the mean to the median as a measure of central tendency for symmetric and skewed distributions to see how they perform. The histograms below allow us to directly compare these two statistics.

With a symmetric distribution, both the mean and the median find the center exactly. They are about the same, and both are valid measures of central tendency.

In a skewed distribution, outliers in the tail pull the mean away from the center toward the longer tail. In this example, the mean differs from the median by more than 9000. The median better represents the central tendency for the skew distribution.

This data is based on US household income for 2006. Income is the classic example of when to use the median rather than the mean because its distribution tends to be skewed. The median indicates that half of all incomes fall below 27581 and half are above. With these data, the mean overestimates where most household incomes fall.

(Video) Math Antics - Mean, Median and Mode

To learn more about income and its right-skewed distribution, read my post onGlobal income distributions.

statisticiansay that the median is a robust statistic, while the mean is sensitive to outliers and skewed distributions.

When to use the median: Skewed Distribution, Continuous Data, Ordinal Data

Related posts:Median Definition and UseandWhat are robust statistics?

## Modus

The mode is the value that occurs most frequently in your dataset, making it a different type of measure of central tendency than the mean or median.

To find the mode, sort the values ​​in your data set by numeric values ​​or by category. Then identify the value that occurs most frequently.

In a bar chart, the mode is the tallest bar. If the data has multiple values ​​that occur most frequently, you have a multimodal distribution. If no value is repeated, the data has no mode. learn more aboutbimodal distributions.

In the following dataset, the value 5 occurs most frequently, making it the mode. This data could represent a 5-point Likert scale.

Typically, you use categorical, ordinal, and discrete data mode. In fact, mode is the only measure of central tendency that you can usecategorical data– like the most popular type of ice cream. However, with categorical data, there is no central value because you cannot order the groups. For ordinal and discrete data, the mode can be an off-center value. Here, too, the mode is the most common value.

On the Quality of Service chart, Very Satisfied is the mode of this distribution because it is the most common value in the data. Notice how it's at the extreme end of the distribution. I'm sure the service providers are happy with these results!

learn more aboutHow to find the mode.

(Video) Mean, median, and mode - measures of central tendency

Related post:Bar Charts: Use, Examples and Interpret

### Find the mode as the central trend for continuous data

In themcontinuous dataNo values ​​are repeated below, indicating that this dataset does not have a mode for a central tendency measure. With continuous data, it is unlikely that two or more values ​​will be exactly the same, since there are infinitely many values ​​between any two values.

If you work with the continuous raw data, don't be surprised if there is no mode. However, you can find the continuous data mode by finding the maximum value on a probability distribution plot. If you canIdentify a probability distribution that fits your data, find the peak and use it as the mode.

The probability distribution plot shows a lognormal distribution with a mode of 16700. This distribution corresponds to the median section of the US household income example.

When to use the mode:Categorical Data, ordinal data, count data, probability distributions

## What is the best measure of central tendency—the mean, median or mode?

If you have a symmetric distribution for continuous data, the mean, median, and mode are the same. In this case, analysts tend to use the mean because it includes all the data in the calculations. However, when you have a skewed distribution, the median is often the best measure of central tendency.

When you have ordinal data, median or mode is usually the best choice. For categorical data, you must use the mode.

In cases where you decide between the mean and the median as a better measure of central tendency, you also determine what types of statistics are usedHypothesentestsare appropriate for your data—if that is your ultimate goal. I wrote an article that discusseswhen to use parametric (mean) and nonparametric (median) hypothesis testingalong with the pros and cons of each type.

Analysts often use measures of central tendency to describe their datasets. learn how toAnalyze descriptive statistics in Excel.

If you are learning about statistics and like the approach I use on my blog, check out mineIntroduction to StatisticsBook! It is available from Amazon and other retailers.

(Video) Mean, Median, and Mode of Grouped Data & Frequency Distribution Tables Statistics

## FAQs

### What are the 4 measures of central tendency? ›

There are three main measures of central tendency:
• mode.
• median.
• mean.

Is mean, median mode best measure of central tendency? ›

Mean is generally considered the best measure of central tendency and the most frequently used one. However, there are some situations where the other measures of central tendency are preferred. Median is preferred to mean[3] when. There are few extreme scores in the distribution. Some scores have undetermined values.

What is the mean, median and mode? ›

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set.

How do you calculate central tendency? ›

The mean is the arithmetic average, and it is probably the measure of central tendency that you are most familiar. Calculating the mean is very simple. You just add up all of the values and divide by the number of observations in your dataset. The calculation of the mean incorporates all values in the data.

What are the 3 measures of central tendency and explain their differences? ›

The 3 most common measures of central tendency are the mean, median and mode. The mode is the most frequent value. The median is the middle number in an ordered data set. The mean is the sum of all values divided by the total number of values.

What does the median tell you? ›

WHAT CAN THE MEDIAN TELL YOU? The median provides a helpful measure of the centre of a dataset. By comparing the median to the mean, you can get an idea of the distribution of a dataset. When the mean and the median are the same, the dataset is more or less evenly distributed from the lowest to highest values.

WHY IS mode the best measure of central tendency? ›

When is the mode the best measure of central tendency? The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency (as it is the only one appropriate to use) when dealing with nominal data.

Why is median the most accurate? ›

The Changing Mean

The value of the mean will change (decrease), but the median will not until a bigger change occurs. Therefore, the median is a more reliable and more stable number than the mean.

What measure of central tendency is the most unreliable? ›

So range is obviously going to be the most unreliable indicator of the floor.

How do you explain mean and mode? ›

The arithmetic mean is found by adding the numbers and dividing the sum by the number of numbers in the list. This is what is most often meant by an average. The median is the middle value in a list ordered from smallest to largest. The mode is the most frequently occurring value on the list.

### What is mode median mean and range and examples? ›

Mean is the average of all of the numbers. Median is the middle number, when in order. Mode is the most common number. Range is the largest number minus the smallest number.

How do I find the median and mode? ›

To find the median, you list your data points in ascending order and then find the middle number. If there are two numbers in the middle, the median is the average of the two. The mode is the most common number in a data set.

What is central tendency with example? ›

Central tendency is a statistic that represents the single value of the entire population or a dataset. Some of the important examples of central tendency include mode, median, arithmetic mean and geometric mean, etc.

What is the formula of mode? ›

In the mode formula,Mode = L+h(fm−f1)(fm−f1)−(fm−f2) L + h ( f m − f 1 ) ( f m − f 1 ) − ( f m − f 2 ) , h refers to the size of the class interval.

How to calculate the median? ›

For a small data set, you first count the number of data points (n) and arrange the data points in increasing order. If the number of data points is uneven, you add 1 to the number of points and divide the results by 2 to get the rank of the data point whose value is the median.

What is the difference between mean and median as measures of central tendency *? ›

In a distribution with an odd number of observations, the median value is the middle value. Advantage of the median: The median is less affected by outliers and skewed data than the mean, and is usually the preferred measure of central tendency when the distribution is not symmetrical.

What is the most commonly used measure of central tendency? ›

Mean is the most commonly used measure of central tendency. There are different types of mean, viz. arithmetic mean, weighted mean, geometric mean (GM) and harmonic mean (HM).

What is central tendency in simple words? ›

Central tendency is a descriptive summary of a dataset through a single value that reflects the center of the data distribution. Along with the variability (dispersion) of a dataset, central tendency is a branch of descriptive statistics. The central tendency is one of the most quintessential concepts in statistics.

What does the mean tell you? ›

The mean and the median are both measures of central tendency that give an indication of the average value of a distribution of figures. The mean is the average of a group of scores. The scores added up and divided by the number of scores. The mean is sensitive to extreme scores when population samples are small.

Why do we use mode? ›

Mode is most useful as a measure of central tendency when examining categorical data, such as models of cars or flavors of soda, for which a mathematical average median value based on ordering can not be calculated.

### What is mode useful for? ›

Advantages of Using Mode

In certain cases, mode can be an extremely helpful measure of central tendency. One of its biggest advantages is that it can be applied to any type of data, whereas both the mean and median cannot be calculated for nominal data.

Where is median used in real life? ›

Median salary helps the employees know the middle point of their salaries in their careers. It is also called the 50 per cent income, which means half of the employees work above this median salary, while half of them work below it.

Why do we use median instead of mode? ›

Because the median only uses one or two values, it's unaffected by extreme outliers or non-symmetric distributions of scores. In contrast, the mean and mode can vary in skewed distributions.

What is better to use mean or median? ›

It's best to use the mean when the distribution of the data values is symmetrical and there are no clear outliers. It's best to use the median when the the distribution of data values is skewed or when there are clear outliers.

What does it mean if mean is higher than median? ›

When the mean is greater than the median, the shape of the distribution is skewed to the right. This means that the bulk of the data are concentrated on the left and there is a long tail stretching to the right.

Which is better median vs mean? ›

The median is a better measure of the central tendency of the group as It it is not skewed by exceptionally high or low characteristic values.

Do you use mean or median for skewed data? ›

For distributions that have outliers or are skewed, the median is often the preferred measure of central tendency because the median is more resistant to outliers than the mean.

What is the easiest measure of central tendency? ›

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

What is the advantage and disadvantage of mean, median mode? ›

The median is less affected by outliers and skewed data. This property makes it a better option than the mean as a measure of central tendency. The mode has an advantage over the median and the mean because it can be computed for both numerical and categorical (non-numerical) data.

How do you define mode in statistics? ›

Mode Definition in Statistics

A mode is defined as the value that has a higher frequency in a given set of values. It is the value that appears the most number of times. Example: In the given set of data: 2, 4, 5, 5, 6, 7, the mode of the data set is 5 since it has appeared in the set twice.

### What is example of mode? ›

Mode: The most frequent number—that is, the number that occurs the highest number of times. Example: The mode of {4 , 2, 4, 3, 2, 2} is 2 because it occurs three times, which is more than any other number.

How do you find the mean? ›

It's obtained by simply dividing the sum of all values in a data set by the number of values. The calculation can be done from raw data or for data aggregated in a frequency table.

What is the mode of this data? ›

The mode of a data set is the number that occurs most frequently in the set. To easily find the mode, put the numbers in order from least to greatest and count how many times each number occurs. The number that occurs the most is the mode! Follow along with this tutorial and see how to find the mode of a set of data.

What are the four measures? ›

You can see that the four levels of measure (nominal, ordinal, interval and ratio) fall into these two larger supercategories.

What is the most commonly used measures of central tendency? ›

Mean is the most commonly used measure of central tendency. There are different types of mean, viz. arithmetic mean, weighted mean, geometric mean (GM) and harmonic mean (HM).

What are the examples of central tendency? ›

For example, if we had four values—4, 10, 12, and 26—the median would be the average of the two middle values, 10 and 12; in this case, 11 is the median. The median may sometimes be a better indicator of central tendency than the mean, especially when there are outliers, or extreme values.

Which of the following is the most commonly used measure of central tendency 4 points? ›

The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. For data from skewed distributions, the median is better than the mean because it isn't influenced by extremely large values.

What are the 3 main types of measures? ›

Three Types of Measures

Use a balanced set of measures for all improvement efforts: outcomes measures, process measures, and balancing measures.

Is Age A nominal or interval? ›

Age can be measured as an interval or a ratio variable. This is because the definition of zero is well defined (no age), the difference between two values is meaningful and the ratio between two values is meaningful as well. In the nominal level of measurement, the variable is categorized but cannot be ranked.

What is ratio vs interval? ›

The difference between interval vs ratio scale comes from their ability to dip below zero. Interval scales hold no true zero and can represent values below zero. For example, you can measure temperatures below 0 degrees Celsius, such as -10 degrees. Ratio variables, on the other hand, never fall below zero.

### What are the uses of mode? ›

Mode is most useful as a measure of central tendency when examining categorical data, such as models of cars or flavors of soda, for which a mathematical average median value based on ordering can not be calculated.

Why median is better than mean? ›

“The mean is typically better when the data follow a symmetric distribution. When the data are skewed, the median is more useful because the mean will be distorted by outliers.”

When would you use the mean? ›

The mean is usually the best measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data is normally distributed. However, it all depends on what you are trying to show from your data.

Why do we use central tendency? ›

Why Is Central Tendency Important? Central tendency is very useful in psychology. It lets us know what is normal or 'average' for a set of data. It also condenses the data set down to one representative value, which is useful when you are working with large amounts of data.

What is the formula of the mode? ›

Mode can be found by using the formula: Mode = 3 Median - 2 Mean.

What is the disadvantage of median? ›

Since the median is an average position, arranging the data in ascending or descending order of magnitude is time-consuming in case of a large number of observations. It is a positional average and does not consider the magnitude of the items. It neglects the extreme values.

What is the advantage of mode? ›

Mode is simple to understand and easy to calculate. It can be located graphically, unlike mean and median. It can be used for qualitative analysis. The extremities in the values of the data do not affect the mode.

## Videos

1. Mean, Median and Mode (Measures of Central Tendency)
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2. Mean, Median, Mode, and Range - How To Find It!
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3. Mean Median and Mode: Understanding and Calculating Measures of Central Tendency
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4. Mean, Median, & Mode - Measures of Central Tendency
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5. What is Central Tendency – An Introduction to Mean, Median, and Mode in Statistics (5-1)
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6. Skewed Distributions and Mean, Median, and Mode (Measures of Central Tendency)
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