Range, Mean, Median, Mode, Outliers and Central Tendency

Range, Mean, Median and Mode of Data

Compute the range, mean, median, and mode of data sets.

Understand the meaning of academic terms: range, mean, median, mode, and data set. The terms range, mean, median and mode are also known as measures of central tendency. They are useful ways of analyzing a set of data. Know that a data set is a list of numbers derived from a collection of data. Know that the range of a data set describes how spread out the data is. Compute the range by subtracting the smallest number from the largest number in the data set. Know that the mean of a data set is the average. Compute the mean of a data set by finding the sum of the values and dividing by the number of addends. Know that the median of a data set is the middle number in the set when that set is arranged in numerical order. Compute the median by first placing all numbers in a data set in order from least to greatest, and then finding the middle number. Know that the mode of a data set is the number that is repeated the most often. Calculate the mode by counting how many of each number to find which number is repeated most often.

It is important to understand the difference between the different measures of central tendency. The meaning of each and the steps needed to calculate them must be memorized. Here are some mnemonic devices to help with the memorization.

Mean: Most Exact Average Number or “It’s mean to add up all the numbers and then divide them again.”

Median: A median in the street is the center divider, it goes down the middle of the road. Likewise, a median in math is in the middle. Also, median, shares sounds with medium. A medium soda is the middle size. The median is the middle number. Visual learners may want to draw this. Median Medium

Mode: The word mode has the same beginning sound as most. The mode is the number you read the most in a data set. Mode Most

Range: Since the range is the difference between the least and the greatest, this visual may help to aid in memorization:

least-RANGE -greatest (the range is between the least and greatest, - stands for difference)

You can also have your child create a picture, a rhyme or song for each measure of central tendency. There is an example of a rap in our links.

(2)

You can find the mean of a number, using two operations (+ then ), in three simple steps.

Add up all the numbers in the data set to calculate the sum.

Count up how many numbers are in the data set. This is your number of addends.

Divide the sum from step 1 by the number of addends from step 2.

Here’s an example of how to use the steps above for the following data set:

2, 4, 5, 6, 2, 4, 5

Add all: 2 + 4 + 5 + 6 + 2 + 4 + 5 = 28, the sum of the addends is 28.

Count the addends, only the numbers next to a + sign. There are 7 addends above.

Divide the sum, 27 by the addends, 7. 28 7 = 4

The mean is 4.

(3)

The median can be found in two steps if the total numbers in your data set are odd and three steps if even. Here’s an example of each.

Odd Data Set: 12, 15, 7, 4, 8

There are a total of 5 numbers in this data set. Since 5 is an odd number, we can find the median in two steps:

Rearrange the numbers so that they are in numerical order from least to greatest. 4, 7, 8, 12, 15

Circle the middle number. This is the median. In the set above, 8 is in the middle of two numbers on each side. So, 8 is the median.

Even Data Set: 11, 15, 9, 15, 5, 10

There are a total of 6 numbers in this data set. Since 6 is an even number, we can find the median in three steps:

Rearrange the numbers so that they are in numerical order from least to greatest. 5, 9, 10, 11, 15, 15

Circle the middle numbers. There will always be two middle numbers if the data set has an even number of data values. In this problem, 10 and 11 are in the middle.

Add the two middle numbers, 10 + 11 = 21. Then divide by 2 to find the average of these numbers, 21 2 = 10.5. The median of this data set is 10.5 or 10 ½. Note, you will always divide by two, because you are finding the average of the two middle numbers.

(4)

The mode(s) of a data set can be calculated by finding the number or numbers that are repeated most often in a data set. To make sure you are counting each number correctly, you can create a table to tally results for each number in the data set. The following table could be used to find the mode for the data set: 2, 3, 2, 4, 2, 4, 3, 3, 2, 3

Number

Tally

Total

2

| | | |

4

3

| | | |

4

4

| |

2

Before choosing using the chart to name the mode(s) make sure you’ve tallied properly. You can check this by totaling your tally marks to make sure they match the total number of items in your data set. The table above has 10 tallies, the data set has 10 numbers, so if tallies were recorded carefully, you’re ready to record the mode(s).

This data set has two modes, 2 and 3, because these numbers are found most often in the data set. Many data sets will only have one mode, however, as you’ve seen above, it is possible to have more than one mode.

(5)

The range tells how far apart the largest number is from the smallest. The range can be calculated in one step. That simple step is; subtract the largest number in the data set from the smallest. So, for the data set: 5, 6, 7, 4, 2, 11.

You would subtract 2 (the smallest number) from 11 (the largest number), as follows;