2. Perhaps the most important measure of location is the
mean.
The mean provides a measure of central location.
The mean of a data set is the average of all the data
values.
The sample mean 𝒙 is the point estimator of the
population mean µ.
Mean
3. 𝑀𝑒𝑎𝑛 =
The arithmetic mean is the most commonly used average.
Definition:
A value obtained by dividing the sum of all observations by their
number, that is
𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒕𝒉𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔
The Arithmetic Mean
4. The population mean is a fixed quantity.
The sample mean is a variable because different samples from the
same population tend to have different mean.
The Arithmetic Mean
5. The mean may correspond to either a population or a sample from the
population.
If the given set of observations represents a population, the mean is called
population mean which is traditionally denoted by 𝝁 (the Greek letter mu). Thus
the population mean of a set of 𝑵 observations 𝒙𝟏, 𝒙𝟐, 𝒙𝟑, … , 𝒙𝑵 is given as
𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 + ⋯ + 𝒙𝑵
𝝁 =
𝑵
𝒙
𝑵
= 𝒊=𝟏 𝒊
𝑵
Where 𝛴, the Greek capital Sigma, is a convenient symbol for summation.
• Ungrouped data
• Population mean
The Arithmetic Mean
6. The Arithmetic Mean
(Ungrouped Data)
If the given set of observations represents a sample, the mean is called a sample
mean, usually denoted by placing a bar over the symbol used to represent the
observations or the variables.
The mean of a set of n observations 𝒙𝟏, 𝒙𝟐, 𝒙𝟑, … , 𝒙𝒏 is defined as:
𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 + ⋯ + 𝒙𝒏
𝒙 =
𝒏
=
𝒏 𝒙𝒊
𝒊=𝟏
𝒏
Where 𝒙 is the mean of a sample of size n.
• Ungrouped data
• Sample mean
7. The Arithmetic Mean
(Example of Population Mean)
Example: (Ungrouped Data)
The number of employees at 5 different drugstores are 3, 5, 6, 4 and 6.
Treating the data as population, find the mean number of employees for
the 5 stores.
Solution: Since the data are considered as finite population,
𝝁 =
3+5+6+4+𝟔
𝟓
= 𝟒. 𝟖
= 𝟓 𝒆𝒎𝒑𝒍𝒐𝒚𝒆𝒆𝒔
8. The Arithmetic Mean
(Example of Sample Mean)
Example: (Ungrouped Data)
The sample of marks obtained by 9 students are given as 45,32,37, 46, 39,
41, 48, 36, 𝟑𝟔. Calculate the
arithmetic mean.
Solution: The mean is given by
=
45 + 32 + 37 + 46 + 39 + 41 + 48 + 36 + 36
𝟗
=
𝟑𝟔𝟎
𝟗
= 𝟒𝟎 𝒎𝒂𝒓𝒌𝒔
11. The Arithmetic Mean
(Grouped Data)
To calculate the mean of grouped data, the first step is to
determine the midpoint of each interval or class. These midpoints
must then be multiplied by the frequencies of the corresponding
classes. The sum of the products divided by the total number of
values will be the value of the mean.
Mathematically,
𝒇𝒊
𝒇𝒊𝒙𝒊
𝒙 = , 𝒊 = 𝟏, 𝟐, 𝟑, … , 𝒏
• Grouped data
• Sample mean
12. The Arithmetic Mean
(Example of Sample Mean)
Example: (Grouped Data)
In Tim's office, there are 25 employees. Each employee travels to work every
morning in his or her own car. The distribution of the driving times (in minutes)
from home to work for the employees is shown in the table below. Calculate the
mean of the driving times.
Driving Times (minutes) Number of Employees
0 to less than 10 3
10 to less than 20 10
20 to less than 30 6
30 to less than 40 4
40 to less than 50 2
13. The Arithmetic Mean
(Example of Sample Mean)
Solution: (Grouped Data)
Driving Times
(minutes)
Number of
Employees (𝒇𝒊)
Mid Points
(𝒙𝒊)
𝒇𝒊𝒙𝒊
𝒇𝒊𝒙𝒊
𝒙 =
𝒇𝒊
𝟓𝟒𝟓
𝒙 =
𝟐𝟓
𝒙 = 𝟐𝟏. 𝟖
0 to less than 10 3 5 15
10 to less than 20 10 15 150
20 to less than 30 6 25 150
30 to less than 40 4 35 140
40 to less than 50 2 45 90
𝒇𝒊 = 𝟐𝟓 𝒇𝒊𝒙𝒊 = 𝟓𝟒𝟓
14. PROPERTIES OF ARITHMETIC MEAN
• Some important properties of the arithmetic mean are as follows:
The sum of deviations of the items from their arithmetic mean is always zero, i.e.
(𝒙 – 𝒙) = 𝟎.
The sum of the squared deviations of the items from Arithmetic Mean (A.M) is
minimum, which is less than the sum of the squared deviations of the items from
any other values.
If we increase or decrease every value of the data set by a specified weight,
then the mean is also increased/decreased by the same digit.
If we multiplied or divided every value of the data set by a specified weight,
then the mean is also multiplied/divided by the same digit.
15. ADVANTAGES OF ARITHMETIC MEAN
Arithmetic mean is simple to understand and easy to calculate.
It is rigidly defined by a mathematical formula.
It is suitable for further algebraic treatment.
It is least affected fluctuation of sampling.
It takes into account all the values in the series.
16. DISADVANTAGES OF ARITHMETIC MEAN
It is highly affected by the presence of a few abnormally high or
abnormally low scores.
In absence of a single item, its value becomes inaccurate.
It can not be determined by inspection.
It gives sometimes fallacious conclusions.
In a highly skewed distribution, the mean is not an appropriate
measure of average.
If the grouped data have “open end” classes, mean can't be
calculated without assuming the limits.
