6th Grade - Mean, Median, Mode & Range

Introduction

• Statistics is a scientific method that uses mathematics to collect data and information for a particular purpose (survey). 
• Then, the collected data is analyzed and interpreted using the measure of central tendency and variability.
• The measure of central tendency is the representative value that would approximately define the entire set of data. It summarizes the data with a single value using mean, mode, and median. 
• The range describes the measure of variability; It reflects the distribution (spread) of the data set. 
• The range can also describe how far the data deviates (off) from the center.

Mean

• The mean ($\overline{x}$) is the most widely used measure of central tendency in which the average of the data is taken. 
• Finding the mean is to determine the fair share. 
• It can be calculated by:

1. 1. Find the sum by adding all the data points.
   2. Divide the sum by the number of all data points (n).

$\overline{x}=\frac{x_1+x_2+...x_n}{n}$

Median

• The median is a measure of central tendency where the mid-value of the data is determined.
• The median of a set of data can be determined by following the steps mentioned below:

1. 1. First, arrange the data set from the least to the greatest.
   2. Then, select the middlemost observation of the data set.
      1. If the total number of observations in the data set is odd, take the middle number as the median.
      2. If the total number of observations in the data set is even, take median will be the average of two middle numbers.

Mode

• Mode is the number that is repeated most frequently in a data set. 
• A set of data can have more than one mode.

Range

•  The steps involved in determining the range of a data set are mentioned below:

1. 1. Find the least (minimum) and the greatest (maximum) number of the data set.
   2. Subtract the minimum (least) value from the maximum (greatest) value.

$Range = The Greatest Value - The Least Value$

Solved Examples

Question 1: The list shows the shoe size of eight children in Mr. Jim's class. 

8, 9.5, 12, 13, 10.5, 11, 12, 12

Find the average children's shoe size in Mr. Jim's class. 

Solution:

1. Find the sum by adding all the data points.

$Sum = 8 + 9.5 + 12 + 13 +10.5 +11 +12 +12 = 88$

2. Divide the sum by the number of all data points.

$\overline{x}=\frac{88}{8}=11$

                  The average shoe size in Mr. Jim's class is size 11. 

Question 2: Find the median of the following data set. 

2, 0, 2, 1, 1, 1, 3

Solution: 

1. Arrange the data set from the least to the greatest. 

$0, 1, 1, 1, 2, 2, 3$ 

2. Determine the median (middle number) of the data set. 

$0, 1, 1, \overline{)1}, 2, 2, 3$

                  Median = 1

 

Question 3: Find the median of the following data set.

2, 0, 3, 1, 1, 4, 5, 7

Solution: 

1. Arrange the data set from the least to the greatest.

0, 1, 1, 2, 3, 4, 5, 7

2. Determined the median (middle) of the data set.

$0, 1, 1, \overline{)2, 3}, 4, 5, 7$

                  Median$=\frac{2+3}{2}=\frac{5}{2}=2.5$

 

Question 4: Find the mode of the following data set:

1, 4, 1, 5, 7, 1, 8

Solution: In the given data set, 1 occurs more often (3 times) than any other number. So, the mode = 1.

 

Question 5: Find the range of the following data set:

100, 120, 88, 89, 150, 110

Solution: 

1. Find the maximum and the minimum value.

$Maximum value = 150$

$Minimum value = 88$

2. Subtract the minimum value from the maximum value. 

$Range = 150 - 88 = 62$

Cheat Sheet

• The three measures of central tendency are mean, median, and mode.
• $\text{Mean}=\frac{\text{sum of all the observations}}{\text{total number of observations}}$
• The Median is the middle-most observation of the given data set.
• Mode is the most frequently occurring observation.
• $\text{Range = Maximum value - Minimum value}$

Blunder Areas

• It is important to arrange the data set from the least to the greatest to find the median.
• A given data set can have more than one mode.