Principles of Statistics 1
Section 4
Example One: The following table shows the number of minutes
required for a student to reach his university within 10 days.
Day One Two Three Four Five Six Seven Eight Nine Ten
Time (Minutes) 39 29 43 52 39 44 40 31 44 35
Calculate the following.
Arithmetic Mean (average)
Median
Mode
Range
Firstly, Calculate Mean
∑𝑛𝑖=1 𝑥𝑖 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛
General rule 𝑥̅ = =
𝑛 𝑛
∑10
𝑖=1 𝑥𝑖 𝑥1 + 𝑥2 + ⋯ + 𝑥10
𝑋̅ = =
10 10
39 + 29 + 43 + 52 + 39 + 44 + 40 + 31 + 44 + 35 396
𝑋̅ = = = 39.6 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
10 10
o Comment: The average time required for the student to reach
the university is 39.6 minutes.
Secondly: Calculate Median
Steps
1. Arranging the data in ascending order
29 31 35 39 39 40 43 44 44 52
2. Find the location of the median
Since the number of observations equal 10 (even number) then,
𝒏 𝒏
𝑶𝒓𝒅𝒆𝒓 𝒐𝒇 𝒎𝒆𝒅𝒊𝒂𝒏 𝒇𝒐𝒓 𝒆𝒗𝒆𝒏 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 = , + 𝟏
𝟐 𝟐
Then,
10 10
𝑂𝑟𝑑𝑒𝑟 𝑜𝑓 𝑚𝑒𝑑𝑖𝑎𝑛 𝑓𝑜𝑟 10 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 = , + 1 = 5,6
2 2
Then,
The value of observation number 5 = 39
The value of observation number 6= 40
3. Calculating the median
The median will be the average of the two middle values
number 5 and number 6
39 + 40
𝑀𝑒𝑑𝑖𝑎𝑛 = = 39.5 𝑚𝑖𝑛𝑢𝑡𝑒
2
Thirdly: calculate the mode and determine its type.
o The mode is the value that appears most frequently then,
The data contain exactly two modes (39, 44) each value repeated
twice.
o The data are bimodal
Fourthly: Calculate the range
Range = largest value – smallest value
Range = 52-29 = 23
Choose the right answer:
From the values of mean, median and mode the data show …….
distribution.
1- Right skewed 2- Symmetric 3- Left skewed 4- No distribution
Example Two: the following table contains the salaries for 7
employees in a company
Employee Salary
Financial Accountant 7500
Head of HR department 5000
Marketer 8000
CEO 68500
Office boy 2500
Security guard 2500
Secretary 4000
Calculate the following.
Arithmetic Mean (average)
Median
Mode
Range
Firstly, Calculate Mean
∑𝑛𝑖=1 𝑥𝑖 𝑥1 + 𝑥2 + ⋯ + 𝑥𝑛
General rule 𝑥̅ = =
𝑛 𝑛
∑10
𝑖=1 𝑥𝑖 𝑥1 + 𝑥2 + ⋯ + 𝑥9
𝑥̅ = =
7 7
7500 + 5000 + 8000 + ⋯ + 4000 98000
𝑥̅ = = = 14000 𝑝𝑜𝑢𝑛𝑑
7 7
o Comment: The average of the salaries in this company is 14000
pound.
Secondly: Calculate Median
Steps
1. Arranging the data in ascending order
2500 2500 4000 5000 7500 8000 68500
2. Find the location of the median
Since the number of observations equal 7 (odd number) then,
𝒏+𝟏
𝑶𝒓𝒅𝒆𝒓 𝒐𝒇 𝒎𝒆𝒅𝒊𝒂𝒏 𝒇𝒐𝒓 𝒐𝒅𝒅 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 =
𝟐
Then,
7+1
𝑂𝑟𝑑𝑒𝑟 𝑜𝑓 𝑚𝑒𝑑𝑖𝑎𝑛 𝑓𝑜𝑟 7 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠 = =4
2
Then,
The value of observation number 4 = 5000
3. Calculating the median
The median will be equal 5000
Thirdly: calculate the mode and determine its type.
o The mode is the value that appears most frequently then,
The data contain exactly one mode (2500). This value repeated
twice.
o The data are unimodal
Fourthly: Calculate the range
Range = largest value – smallest value
Range = 68500 - 2500 = 66000
Choose the right answer:
From the values of mean, median and mode the data show …….
distribution.
1- Right skewed 2- Symmetric 3- Left skewed 4- No distribution
Example Three: Find the mode in the following data
Dataset one
10 1 1 2 2 2 0 0 0 1
Dataset Two
10 6 22 55 88 43 89 5 25 27
The solution
Dataset Number The answer
Dataset One 0,1,2
Dataset Two No mode
Question Five: Given the following Macroeconomics exam
grades for 8 students.
81 75 62 16 84 96 66 58
a. Calculate the range
b. Calculate the variance and standard deviation for this dataset.
∑(𝑥−𝑥)2
- The variance of the sample 𝑆 2 =
𝑛−1
- The standard deviation of the sample 𝑆 = √𝑆 2 =
∑(𝑥−𝑥)2
√
𝑛−1
In our example we have a sample consist of 8 students so we
will use the sample rule for the variance and standard deviation
𝑥 𝑥 𝑥−𝑥 (𝑥 − 𝑥)2
16 -51.25 2626.5625
67.25
58 67.25 -9.25 85.5625
62 67.25 -5.25 27.5625
66 67.25 -1.25 1.5625
75 67.25 7.75 60.0625
81 67.25 13.75 189.0625
84 67.25 16.75 280.5625
96 67.25 28.75 826.5625
Sum 538 0 4097.5
16+58+62+66+75+81+84+96
- Mean = = 67.25
8
4097.5
- The variance of the sample 𝑆 2 = = 585.357
7
- The standard deviation of the sample 𝑆 = √585.357 =
24.194
|∑ 𝑋−𝑋̅| 134
- The Absolute Deviation Mean = = = 16.75
𝑛 8
|∑ 𝑋 − 𝑋̅| = 51.25 + 9.25 + 5.25 + 1.25 + 7.75 + 13.75 + 16.75 + 28.75
= 134
