Difference Between Sample Mean and Population Mean
Sample Mean vs 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.”
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.
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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