Population Mean Formula

Population Mean Formula (Table of Contents)

Population Mean Formula
Examples of Population Mean Formula (With Excel Template)
Population Mean Formula Calculator

In statistic, the population is basically a collection of a group of things. This can be of numbers, people, objects, etc. So the population means is nothing but the average of this group of items. It is basically arithmetic mean of the group and can be calculated by taking a sum of all the data points and then dividing it by the number of items we have in the group. It is the most common method to measure the center of a data set but it is very rare that we calculate population mean. The reason for that is population is a big data set and it is very time-consuming and costly to find the population mean. For example, the age of people living in Washington DC is the population set; it is very difficult to count each person and then take an average. So usually what we do is that we extract a sample out of the population which is a representation of population set and we take an average of a sample of seeing what is the average of the population.

A formula for Population Mean is given by:

Population Mean = Sum of All the Items / Number of Items

In case you want to use the sample mean as representative of the population mean:

Sample Mean = Sum of All the Items in Sample / (Number of Items in Sample – 1)

Examples of Population Mean Formula (With Excel Template)

Let’s take an example to understand the calculation of Population Mean formula in a better manner.

You can download this Population Mean Template here – Population Mean Template

Example #1

Let say you have a data set with 10 data points and we want to calculate Population Mean for that.

Data set : {14,61,83,92,2,8,48,25,71,12}

Solution:

Population Mean is calculated using the formula given below

Population Mean = Sum of All the Items / Number of Items

Population Mean = (14+61+83+92+2+8+48+25+71+12) / 10
Population Mean = 416 / 10
Population Mean = 41.6

Example #2

Let say you want to invest in IBM and very keen to look at its past performance and returns. You want to go back 20 years and calculate monthly return but that will become very hectic. So you have decided to take a sample of the last 10 months and calculate return and mean of that. You believe that the sample you have taken is a correct representation of the population.

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Solution:

So if you see here, in the last 10 months, IBM return has fluctuated very much.

Sample Mean is calculated using the formula given below

Sample Mean = Sum of All the Items in Sample / (Number of Items in Sample – 1)

Sample Mean = (3.74% + 1.07% +4.34% + (-23.66)% + 7.66% + (-7.36)% + 18.25% + 2.76% + 1.48% + 0.00%) / (10 – 1)
Sample Mean = 8.28% / 9
Sample Mean = 0.92%

Overall, in the last 10 months, the average return is only 0.92%.

Explanation

Mean, in general, is a simple average of the data points we have in a data set and it helps us to understand the average point of the data set. But there are certain limitations of using mean. These limitations are valid for both Population and Sample Mean. First of all, the mean value is easily distorted by extreme values. For example: Let say we have returns of stock for the last 5 years given by 5%, 2%, 1%, 5%, -30%. Mean for these values is -3.4% ((5+2+1+5-30)/5). So although the stock has provided a positive return for the first 4 years, on an average we have a negative mean of 3.4%. Similarly, if we have a project for which we are analyzing the cash flow for the next 5 years. Let say the cash flows are: -100, -100, -100, -100, +1000. Mean is 600 / 5 = 120. Although we have a positive mean, we are only getting money in last year of the project and it can happen that if we incorporate time value of money, this project will not look as lucrative as it is now.

Relevance and Uses of Population Mean Formula

In general, Population Mean is very simple yet one of the crucial elements of statistics. It is the basic foundation of statistical analysis of data. It is very easy to calculate and easy to understand also. But as mentioned above, the population mean is very difficult to calculate, so it is more of a theoretical concept. It does not make sense to spend enormous efforts to find a mean of population set. So sample mean is a more realistic and practical concept. Also, mean value, if look it in a silo, has relatively less significance because of the flaws discussed above and it is more of a theoretical number. So we should use mean value very carefully and should not analyze the data only based on the mean.