Sample mean formula & Solved Examples

A sample is a cluster or group of elements chosen from a population. It helps to estimate what the entire population is doing, without having to survey everyone. Hence, it is a method of saving plenty of time and money. The attributes used to define a population are referred to as the parameters and the properties of a sample data are known as statistics. The concept of both the population and sample are essential components of statistics. A sample statistic is a portion of the data collected from a fraction of a population. It should define the population as a whole and not reflect any inclination toward a specific attribute. The concept of samples is being used for studies and measurements by scientists, marketers, government agencies, economists, research groups, etc. It involves important topics such as hypothesized mean, mean standard deviation, distribution of means, and various other concepts.

Key takeaway: Sample, mean, population, attributes, statistics, cluster, hypothesized mean, standard deviation, distribution, unbiased, observations, median, frequency, sampling

What is sample in statistics?

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A sample is generally a smaller and manageable version of a larger group. It is a subset having the attributes of a larger population. A population, in general, is the total number of observations contained in a given group or context. When the sizes of the population are huge to include all possible members or observations, then the concept of sampling is used in statistical testing. 

The image below shows the relation between a population and the sample. 

Sample of a Population

As shown in the above figure, a sample is an unbiased number of observations carried out from a population. In other words, it is a portion or fraction of the whole group and functions as the subset of a population. 

Sample mean

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In statistics, the mean is termed as one of the measures of central tendency. It is similar to the average of a given set of values. It represents the equal value distribution for the data set provided. It is another term for “median.” Generally, the sample mean is an average sample size and is only a tiny component of the total population. It is helpful as it helps to evaluate what is being done by the entire population without conducting any survey. 

Sample Mean formula

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The sample mean of any population can be determined by adding all the observations and then dividing the resultant by the total number of observations made i.e. ‘n’. The formula for evaluating the sample mean of a population is given as:

\(\bar{X}\) = (\(\sum\) x_i) / n or \(\bar{X}\) = 1/ n * (\(\sum\)x_i)

From the above equation, it can be concluded that the concept of average and sample mean is nearly the same. However, the only difference existing between both of them is their symbols. 

In the above equation, 

\(\bar{X}\) = stands for ‘sample mean’

\(\sum\)= summation notation

x_i ⁿ = all of the values of x

n = number of items in sample mean

Hypothesized mean

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In statistics, hypothesis testing is a technique in which an analyst tests an assumption or a hypothesis about a population parameter. It is used to evaluate the probability of a hypothesis by using sample data that may have come from a larger population or any data-generating process. 

Its process involves the use of a random population sample for testing two different hypotheses which are: 

• Null hypothesis (H₀)
• Alternative hypothesis (H_i)

The null hypothesis is the hypothesis of equality between population parameters whereas the alternative hypothesis is the opposite of a null hypothesis.

Estimating the mean

For calculating the mean of a given population or group, follow the steps listed below:

Step 1: First create a new column into the table and write the middle value (midpoint) of each group.

Step 2: Now multiply middle values by the frequency of that group and Step 3: Create a new column for (midpoint x frequency) values and add the results obtained from the above step into the column.

Step 4: At last, divide that value obtained by the net frequency to obtain the mean.

Sampling distribution

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A sampling distribution also referred to as a finite-sample distribution, is similar to the probability distribution of a statistic that is selected from random samples of a population. It illustrates the distribution of frequencies for how to spread apart various outcomes for a specific population.

It relies on numerous aspects like statistics, sample size, sampling process, the overall population, etc. and helps in calculating the statistics such as means, ranges, variances, and standard deviations for the sample provided.

The variability of a sampling distribution is determined by the number of observations in a population and a sample, and the approach used to draw the sample sets.

Solved examples

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Example 1: An exam of 100 marks was conducted in Class 10th of 30 students and marks obtained by them have been recorded in the table below. 

You have to calculate the mean of the obtained marks by the students. 

Solution: We will take the marks obtained by the students as xi and number of students as fi. We will then calculate the product of fixi for each observation individually. 

Using the formula for mean we will now calculate the mean of marks obtained by the students:

Mean,\(\bar{X}\) = \(\sum f_i. x_i/ \sum f_i\)

=> 1779/30

=> 59.3

Hence the mean marks obtained is 59.3 

Example: A survey was conducted in a locality of 25 households for daily expenditure on food. The table below contains the details of the survey. You have to evaluate the daily expenditure on food by an appropriate method. 

Solution: From the given question, we can take the value of assumed mean a, and class interval h as 225 and 50 respectively.

Mean, \(\bar{X}\) = a + h * (\(\sum f_i u_i* \sum f_i\))

Putting the values of a and h in the above equation, 

\(\bar{X}\)= 225 + 50 * (-7/25)

=> 225 -14

=> 211

Hence, main daily expenditure on food calculate will be Rs. 211

Things to remember

• The sample mean is referred to as the average value found in a sample. It is simply a small portion of the whole population. In other words, mean refers to “average.”
• Sample mean is determined using the formula \(\bar{X}\) = 1/ n * (Σ x_i) where n is the total number of observations. 
• Hypothesis testing is an important approach in statistics used to evaluate two mutually exclusive statements about a population that specify which statement is most suitable and supported by the sample data.
• A sampling distribution is typically a graph of a statistic for the sample data. It’s the probability distribution of a statistic achieved from a larger number of samples taken from a specific population. For a given population, the sampling distribution is the allocation of frequencies of a range of distinct outcomes that could occur for a statistic of a population.