x̄ which is read as x bar is a fundamental concept for understanding and interpreting data in Statistics. x̄ also called as sample mean is a measure of central tendency i.e. the average value of given sample data points with a single value.

In this article, we are going to learn what is x̄, how we can calculate x̄, the relationship of x with other statistical terms, and some practice questions on x̄.

Definition of x̄

x̄ is a measure of central tendency that represents the average value of given sample data points with a single value. x̄ is also known as a sample mean. Mathematically, it is defined as the sum of all given data points divided by the total number of data points. The formula for calculating x̄ or sample mean is given below:

x̄ = ( x₁ + x₂ + x₃ +……….+ x_n)/n

where,

• x₁, x₂, ….x_n represents each data point in the sample
• n represents the total number of data points in the sample

When we deal with population mean then x̄ is denoted by μ and calculates the same as this, but for the entire population data set.

How to Calculate x̄

For a given sample of data

To calculate the sample mean x̄. Follow the steps given below:

Step 1: Sum all the data points given in the sample.

Step 2: Divide the sum by the total number of data points.

Example: For given a sample data set {4, 8, 6, 5, 3, 7}, Find x̄.

Solution:

x̄ = (4 + 8 + 6 + 5 + 3 + 7)/6

x̄ = 33/6

x̄ = 5.5

For Population

The calculation for the population mean μ is similar to the x sample mean but it includes all data points in the population.

Example: Calculate μ for a given population data set: {3, 6, 2, 4, 5, 7, 1}

Solution:

μ = (3 + 6 + 2 + 4 + 5 + 7 + 1)/7

μ = 28/7

μ = 4

Relationship with Other Statistical Measures

Variance and Standard Deviation

The mean is related to the variance and standard deviation, which measure the spread of the data around the mean. Variance is the average of the squared differences from the mean, and the standard deviation is the square root of the variance. Variance and Standard Deviation tells us about the change of the data.

Variance (s² or σ²)

s² = Σ_{i = 1}ⁿ(x_i – x)² / (n-1 ) for a sample

σ² = Σ_{i =1} ⁿ(xi – μ)² / (n-1 ) for a population

Importance of x̄ in Statistics

Importance of x̄ in Statistics is studied under two headings:

• Descriptive Statistics: x̄ is used to summarize the central tendency of a data set in descriptive statistics. It provides the understanding and comparison of various data sets by providing a single value that represents the center of the data distribution.

• Inferential Statistics: x̄ is used to make inferences about a population based on a sample in inferential statistics. It is important to many statistical tests and computations of confidence intervals, which helps researchers to conclude population parameters.

Misconceptions About x̄

Some misconception about x̄ are:

• Sometimes the mean is often confused with the median. The mean is the average of all given data points in the sample, while the median is the middle value when the data points are arranged in ascending order.

• Mean is sensitive to extreme values, which can distort the results. In contrast, the median is more robust to outliers.

Real-Life Application of x̄ Bar in Statistics

Various application of x̄ bar in Statistics includes:

• To determine the effectiveness and safety of new treatments or medications, clinical trials are used in healthcare.

• As far as research in education is concerned, researchers in education employ the sample mean to study the data coming from the researches done on teaching methods, learning outcomes, and educational interventions.

• The manufacturers use the sample mean to follow up on the progress and to improve production processes.

• The sample mean from a sample of ampoules is used by quality assurance teams to inspect data from controls and tests.

• Businesses conduct market investigations to gain deeper insight into where consumer preferences lie, which may, in turn, explain behavior.

Solved Examples on x̄ in statistics

Examples 1: For given sample data: {3, 7, 5, 10, 2}. Calculate the x̄ i.e. sample mean.

Solution:

For calculating sample mean, we will use the formula:

x̄ = ( x₁ + x₂ + x₃ +……….+ x_n)/n

Sample Mean = (Sum all Data Points)/(Number of Data Points)

x̄ = (3 + 7 + 5 + 10 + 2)/5

= 27/5

= 5.4

So, sample mean (x̄) is 5.4

Examples 2: Calculate the weighted mean of the following data: (3, 0.2) , (7, 0.3) , (5, 0.5).

Solution:

Weighted mean can be calculated by multiplying each value by its weight and summing these values, then dividing by the total sum of the weights.

x̄_w = (( 3 × 0.2) + (7 × 0.3) + (5 × 0.5))/(0.2 + 0.3 + 0.5)

= (0.6 + 2.1 + 2.5)/1.0

= 5.2

So, the weighted sample is 5.2

Examples 3: Given two samples, Sample A: {5, 7, 9, 11, 13} and Sample B: {4,6,8,10,12}, compare their sample means.

Solution:

First we calculate sample mean for Sample A data :

x̄_a = (5 + 7 + 9 + 11 + 13)/5

= 45/9

= 5

Now we calculate sample mean for Sample B data :

x̄_b = 4 + 6 + 8 + 10 + 12/5

= 40/5

= 8

From both sample mean we can observe that Sample A has a higher sample mean as compared to Sample B.

Examples 4: Calculate the sample mean for the data set {4, 6, 8, 10, 100} and explain the impact of the outlier.

Solution:

First we calculate the sample mean of given data

x̄ = (4 + 6 + 8 + 10 + 100)/5

x̄ = 128/5

= 25.6

From given data we can observe that outline is 100 and it significantly increases the sample mean from what it would be if the data were more uniformly distributed.

Examples 5: Given the frequency table below, calculate the sample mean.