Mean, Median & Mode
- Simple statistical analysis may include calculating the average of a given set of numerical data, using one of three methods
- The mean is commonly considered the true average - where all the numbers in a data set are added and then divided by the number of numbers
- The median is the middle value in the list of numbers
- The mode is the value that occurs most often in a set of data
Worked example
RapidKleen kept a record of mobile vehicle valets carried out each day during a busy holiday period.
Find the mean, median, mode, and range of mobile valets during the period using the following data.
Day |
|
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
No. of valets sold |
13 |
18 |
13 |
14 |
13 |
16 |
14 |
21 |
13 |
[6 marks]
Step 1: To calculate the mean, first add together each of the values
13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13 = 135
Step 2: Divide the total by the number of values
135 ÷ 9 = 15
[2 marks]
- Note that the mean in this case isn't a value from the original data set
- This is a common result - you should not assume that your mean will be one of your original numbers and you should not be surprised when it isn't
Step 3: To calculate the median first rewrite the data set in numerical order
13, 13, 13, 13, 14, 14, 16, 18, 21
Step 4: Identify the middle number
- There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:
So the median is 14
[2 marks]
- Note: The formula for the place to find the median is "([the number of data points] + 1) ÷ 2", but you don't have to use this formula
- You can just count in from both ends of the list until you meet in the middle if you prefer, especially if the data set is small
- You can just count in from both ends of the list until you meet in the middle if you prefer, especially if the data set is small
Step 5: To calculate the mode rewrite the data set in numerical order
13, 13, 13, 13, 14, 14, 16, 18, 21
Step 6: Identify the number that occurs most often in the list
13 occurs four times
14 occurs twice
16 appears once
18 appears once
21 appears once
As it appears most frequently the mode number of valets sold is 13
[2 marks]
Standard Deviation
- The standard deviation is a measure of the spread of numbers within a set of data
- It is a particularly useful tool for planning when managers have wide ranges of data and need to organise resources effectively
Worked example
FreshBite is a pre-packaged sandwich manufacturer which produces a range of products that are sold in cafés and refreshment stands in tourist attractions such as theme parks.
Freshbite's sales are highly variable - the business regularly suffers from high levels of wastage as a result of having large quantities of unsold stock. On several occasions it has also been unable to fulfill orders from customers as it has not produced enough units.
The business has recently employed a new operations manager who has suggested that calculating the standard deviation of sales would aid planning. He has requested the last month's sales data to allow him to calculate this.
Product |
Last month's sales ($) |
A | 110,000 |
B | 27,000 |
C | 12,000 |
D | 54,000 |
E | 7,000 |
Calculate the standard deviation of last months' sales for Freshbite.
[4 marks]
Step 1: Calculate the mean
110,000 + 27,000 + 12,000 + 54,000 + 6,000 = 210,000
210,000 ÷ 5 = 42,000
[1 mark]
Step 2: For each product, subtract the mean and square the result
Product |
Last month's sales ($000s) |
|
|
A | 110 | 68 | 4,624 |
B | 27 | -15 | 225 |
C | 12 | -30 | 900 |
D | 54 | 12 | 144 |
E | 7 | -35 | 1,225 |
[1 mark]
Step 3: Add up the squared differences and express in an expanded form
4,624 + 225 + 900 + 144 + 1,225 = 7,118
= 7,118,000
[1 mark]
Step 4: Find the square root to identify the standard deviation
[1 mark]
Note - in this instance, a significant standard deviation from the mean informs Freshbite's managers that they need to carefully plan for significant variations in sales. This may include detailed market research as well as capital investment to reduce wastage (for example, further freezers).