Unit E: Statistics and Probability

Chapter 1: Statistics

Weighted Mean

Jordan plans to visit Cancun, Mexico, and records the rainfall for the last seven days in a table.

Day	Amount of Rainfall (mm)
Sunday	1
Monday	6
Tuesday	5
Wednesday	5
Thursday	5
Friday	3
Saturday	6

He uses the mean formula to calculate the average rainfall over the last seven days.

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7}}{n} \\ = & \frac{1 + 6 + 5 + 5 + 5 + 3 + 6}{7} \\ = & \frac{31}{7} \\ = & 4.4 \end{array}

where
x₁ is the first value in the data set
x₂ is the second value in the data set
x₃ is the third value in the data set
x_n is the last value in the data set
n is the number of values in the data set

He calculates the mean rainfall to be 4.4 mm.

Jordan's sister, Tina, notices that only four values appear in the table: 1, 3, 5, and 6. She rewrites the table to group the data.

Amount of Rainfall (mm)	Number of Days
1	1
3	1
5	3
6	2

Tina explains that the weighted mean formula is easier to use to calculate the mean when certain values are worth more or appear more than others. The weighted mean is an average in which each value is assigned a weight depending on its frequency, or how often the value appears in a set of data. Tina calculates the mean rainfall using the weighted mean formula.

\begin{array}{rcl} \bar{x} & = & \frac{sum of the values in the data set}{total number of values in the data set} \\ = & \frac{w_{1} (x_{1}) + w_{2} (x_{2}) + w_{3} (x_{3}) + \dots + w_{n} (x_{n})}{w_{1} + w_{2} + w_{3} + \dots + w_{n}} \end{array}

where
w₁ is the weight of the first value in the data set
w₂ is the weight of the second value in the data set
w₃ is the weight of the third value in the data set
w_n is the weight of the last value in the data set
n is the number of values in the data set

Tina calculated the mean rainfall as

\begin{array}{rcl} \bar{x} & = & \frac{w_{1} (x_{1}) + w_{2} (x_{2}) + w_{3} (x_{3}) + w_{4} (x_{4})}{w_{1} + w_{2} + w_{3} + w_{4}} \\ = & \frac{1 (1) + 1 (3) + 3 (5) + 2 (6)}{1 + 1 + 3 + 2} \\ = & \frac{1 + 3 + 15 + 12}{1 + 1 + 3 + 2} \\ = & \frac{31}{7} \\ = & 4.4 \end{array}

Tina calculated the mean rainfall to be 4.4 mm, which is equal to Jordan's answer.

Both the mean and weighted mean can be used to calculate the mean. The weighted mean formula should be used instead of the mean formula if

the data values are repeated and it is easier to calculate using the weighted mean formula
OR
the data is grouped
OR
the values in the set of data are weighted differently