## Table of Contents

- What are the Measures of Central Tendency?
- What are Mean, Median, Mode, and Range Used for?
- Example Using Mean, Median, Mode, and Range
- Lesson Summary

What are the measures of central tendency? Learn how to find the mean, median, mode and range in a data set, how each is used in math and view examples.
Updated: 06/21/2021

- What are the Measures of Central Tendency?
- What are Mean, Median, Mode, and Range Used for?
- Example Using Mean, Median, Mode, and Range
- Lesson Summary

"Measures of Central Tendency" is one of the most important concepts in the study of **descriptive statistics**, a type of statistics used to describe and summarize values in a data set. Measures of central tendency do not provide information about the individual values of a data set. Rather, it is a set of numerical values that best represent or summarize the middle of the data set. The following measures can be used to determine the central tendency:

**Mean**: the average of the individual values of a data set.**Median**: the number in the middle of a data set with an equal number of values higher and lower than it.**Mode**: the number or value that occurs the most in a data set.

Range is often included when finding the measures of central tendency. However, the range is actually a **measure of variation**. A measure of variation describes the variability of the data-- or how spread out the data is in a data set.

The **mean** does not always provide the best measure of central tendency for a data set. When the data is in a **symmetrical distribution**, or bell-curve, the mean, median, and mode are all at the center of the distribution. If a data set contains **outliers**, or values that are significantly higher or lower than the rest of the values in the data set, it can look skewed when represented on a graph. A **skewed distribution **is when the tail of a distribution on a graph is longer than the other. If the tail is longer towards the negative side of a number line, it is left-skewed. If the tail is longer on the positive side of a number line, it is right-skewed. The more a distribution is skewed, either right or left, the less accurate the mean.

Unlike the mean, the **median** is not calculated using all of the numbers in the data set, so the median is not impacted by extreme **outliers** or heavily skewed distributions.

The **mode** provides the value that occurs most frequently within a data set. The **mode** is frequently used with data sets that include **categorical data**, or data that can be organized into groups, but does not have mathematical meaning. Zip codes and phone numbers are types of numerical data that do not have a mathematical meaning because they do not indicate trends in a data set. Mode is useful for finding the most popular option in a categorical data set like the most favorite color.

**Range** is a numerical value that represents the spread of a data set. The range can be misleading if there are extreme outliers in the data set.

In mathematics, finding the mean is synonymous with finding the average of a set of numbers.

To calculate the mean:

- Determine the total number of values within the dataset.
- Determine the sum of the values in the data set.
- Divide the sum of all the values by the total number of values in the dataset.

There are two ways to find the median of a data set. The correct method depends on the number of values within the set.

If the data set has an odd quantity of numbers in the set:

- Organize the values in the set "by magnitude" (from least to greatest)
- Find the value that is located directly in the middle of the set.

If the data set has an even quantity of numbers in the set:

- Organize the values in the set by magnitude.
- Find the two values that are located directly in the middle of the set.
- Add the the two values together then divide the sum by two.

The mode is the number that occurs the most in the data set. The mode may not be a unique value if no single number was repeated more than others.

- Organize the data by groups or magnitude.
- Identify the value that occurs most frequently within the set.

The range is the numerical difference between the maximum and minimum.

- Identify the maximum (largest) and minimum (smallest) values of a data set.
- Subtract the minimum from the maximum.

The following is an example of how the mean, median, mode, and range can all be drawn from the same data set.

American Tiger University recorded the following ages for their 2021-2022 academic year roster:

Student's Name | Age |

Mindy | 19 |

Cameron | 22 |

Tony | 19 |

Keith | 23 |

Lauren | 19 |

Tommy | 21 |

Pilar | 25 |

Mike | 19 |

Carla | 27 |

Kira | 25 |

Marcus | 21 |

To find the mean of the American Tiger University roster, or "data set,":

- Find the sum of the values of the set.
- Divide the sum by number of values in the set.

$$\large \frac{19+22+19+23+19+21+25+19+27+25+21}{11} = \frac{240}{11}=21.\bar{81} $$

The average age of a member on the swim team is approximately 21.81.

To find the median of the data set:

- Organize the ages from least to greatest.
- This set currently has an odd set of values (ages), there will be one median.

21 is the single number directly in the center, and the median of the swim team is therefore 21.

Jordan is eighteen years old and freshman who's just joined the American Tiger University swim team, and the roster has changed. Now the second method for finding the median is necessary. Now to find the median:

- With the new value (age) added to the data set, there is a now an even number of values.
- Find the average of the two middle values.

3. To find a single value for the median, calculate the average of these two values. The median age of a member on the swim team is 21 years old.

$$\large \frac{21 + 21}{2}= 21\ $$

The value (age) 19 occurs four times and most often in the data set. The most common age of the team members is 19 years. The mode is therefore 19.

The graph below demonstrates the distribution of the data set. The distribution is slightly skewed right ("positive skewness").

- This distribution includes Jordan's age.
- The new mean including Jordan's age is 21.5 years.
- The median is 21 years.

This distribution is slightly skewed right and indicates that the mean is only slightly greater than the median. The mean is not heavily impacted by a possible outlier.

For this data set, the mode (19 years) is the lowest measure of central tendency and the mean (21.5 years) is the highest. The median (21 years) is the value in the middle. Since the mean and median only differ by 0.5 years, both could be considered the best representation of this data set.

To find the range of the data set:

- Identify the minimum value and the maximum value of the data set.
- Subtract the minimum value from the maximum value.

$$\large 27 - 18 = 9 $$

The range of ages of the members on the swim team is 9 years.

This lesson covers the following regarding how to find measures of central tendency:

- Measures of central tendency are a set of numerical values that best represent or summarize the middle of the data set.
- Mean, also as the average, can be found by finding the sum of the values and dividing that sum by the number of values in the set.
- Median is the number in the middle of a data set with an equal number of values higher and lower than it.
- Mode is the value that occurs the most within a set, there can be more than one mode or no mode.
- Range is a measure of variation that indicates the spread of the data. The range can be found by subtracting the minimum value from the maximum value.
- In a symmetrical distribution, mean, median, and mode are at the center of the distribution.
- In a heavily skewed distribution, the median (and in some cases the mode) may provide a more accurate representation of the data.

This lesson has covered the following vocabulary words:

**Descriptive statistics:**a type of statistics used to describe and summarize values in a data set**Categorical data**: a type of data that can be organized into groups, but does not have mathematical meaning.**Measure of variation:**a description of the variability or how spread out the data is in a data set.**Symmetrical distribution:**when the data creates a symmetrical bell-curve on a graph.**Outliers:**A value that is significantly higher or lower than the rest of the values in the data set**Skewed distribution:**When the tail of a distribution is longer than the other. If the tail is longer towards the negative side of a number line, it is left-skewed. If the tail is longer on the positive side of a number line, it is right-skewed.

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Frequently Asked Questions

In mathematics the mean is considered to be the same thing as the average. Both are calculated by finding the sum of the total values in the set, then dividing that number by the number of values within the set.

Range is a measure of variability and represents the spread of the values in a data set. Mode is the number or value that occurs most often in the data set, or the most popular value. Mean is the average, or center of a normal distribution. The mean may not be as accurate in data sets with outliers or skewed distributions.

The mean of a data set can be found by adding the values together and dividing that sum by the total number of values in the set. The median can be found by organizing the values from least to greatest and locating the value that is directly in the center of the set. If there are two values, the mean can be found by adding those two values and dividing that sum by two. The mode is the value that occurs most frequently within a set of data.

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