Difference Between Sample Mean and Population Mean

Sample Mean vs Population Mean

“Mean” is the average of all the values in a sample. It can be calculated by adding up all the values and then dividing the sum total by the number of values in the sample.

Population Mean
When the provided list represents a statistical population, then the mean is called the population mean. It is usually denoted by the letter “µ.”

Sample Mean
When the provided list represents a statistical sample, then the mean is called the sample mean. The sample mean is denoted by “X.” It is a satisfactory estimate of the population mean.
For a sample, a population mean may be defined as:
µ = Σ x / n where;

Σ represents the sum of all the number of observations in the population;
n represents the number of observations taken for the study.

When frequency is also included in the data, then the mean may be calculated as:
µ = Σ f x / n where;

f represents the class frequency;
x represents class value;
n represents the size of the population, and
Σ represents the summation of the products “f” with “x” all over the classes.

In the same way the sample mean will be;
X = Σ x / n or
µ = Σ f x / n where “n” is the number of observations.
In a more elaborate way it may be represented as;
X = x₁ + x₂ + x₃ +…………….xn / n or
X = 1/n(x₁ + x₂ + x₃ +…………….xn ) = Σ x / n
This can be cleared with the following example:
Suppose the data has the following observations of a study.
1, 2, 2, 3, 3, 4, 5, 6, 7, 8
For these samples to take out the sample mean, we will consider several samples and consider the mean.
For 1, 2, 3, mean will be calculated as (1+ 2+3/ 3) = 2;
For 3, 4, 5, mean will be calculated as (3 +4 + 5/3) = 4;
For 4, 5, 6, 7, 8, mean will be calculated as (4 +5+6 +7 +8/5) = 6;
And for 3, 3, 4, 5, mean will be calculated as (3 + 3 +4 + 5/4) = 3.75.
Thus the total mean of these samples is (2 + 4+ 6 + 3.75/ 4) = 3.94 or approximately 4.
This value is called the sample mean.
Now for the population, the population mean can be calculated as:
1+ 2+ 2+ 3+ 3+4+5+ 6+7+ 8/10 = 4.1
Thus the sample mean is very close to the population mean. The accuracy increases with an increase in the number of samples taken.

Summary:

1.A sample mean is the mean of the statistical samples while a population mean is the mean of the total population.
2.The sample mean provides an estimate of the population mean.
3.A sample mean is more manageable data while a population mean is difficult to calculate.
4.The sample mean increases its accuracy to the population mean with the increased number of observations.

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