Mean of Combined Distributions and Combined Standard Deviation

Mean of Combined Distributions and Combined Standard Deviation: When we have two or more data points, finding the average of total data is possible by combining the mean of the distribution. The combined mean of the distributions can be used in daily life applications such as finding the mean of a class and mean wages.

Variance is the measure of distribution from the mean, and the square root of the variance gives the standard deviation of the data. The combined variance and combined standard deviation are found by adding the variance and standard deviation of each data group, separately. Probability distribution is the study of the total possible outcomes of an event.

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Mean of Combined Distributions and Combined Standard Deviation: Combined Mean

A combined mean is the mean of two or more separate groups. It is calculated by calculating the mean of each group and combining (adding) the results of the individual means.

We can calculate the mean of combined distributions data if we know the mean and the number of observations in each group of data, and it is calculated by using the formula:

\(x_{c}=x_{12}=\frac{n_{1} \cdot \overline{x_{1}}+n_{2} \cdot \overline{x_{2}}}{n_{1}+n_{2}}\)

Where,

• \(\overline{x_{1}}=\) mean of the first set
• \(n_{1}=\) number of items in the first set
• \(\overline{x_{2}}=\) mean of the second set
• \(n_{2}=\) number of items in the second set
• \(x_{c}=\) combined mean

A combined mean is just a weighted mean in which the weights are the group sizes.

When there are more than two groups:

Step 1: Add the means of each group—each weighted by the number of individuals or data points—and divide by the total number of data points.

Step 2: Divide the result of Step \(1\) by the total number of people (or data points).

\(x_{12}=\frac{n_{1} \cdot \overline{x_{1}}+n_{2} \cdot \overline{x_{2}}+n_{3} \cdot \overline{x_{3}}+\cdots}{n_{1}+n_{2}+n_{3}+\cdots}\)

Where,

• \(\overline{x_{1}}=\) mean of the first set
• \(n_{1}=\) number of items in the first set
• \(\overline{x_{2}}=\) mean of the second set
• \(n_{2}=\) number of items in the third set
• \(\overline{x_{3}}=\) mean of the third set
• \(n_{3}=\) number of items in the first set
• \(x_{c}=\) combined mean

Variance

The variance measures how far a set of data is distributed from its mean or average value. Variance is denoted by the symbol ‘ \(\sigma^{2}\) ‘.’

Combined Standard Variance

The combined variance or standard deviation, like the combined mean, can be calculated for different sets of data.

Assuming we have two sets of data with \(n_{1}\) and \(n_{2}\) observations, respectively, with averages \(X_{1}\) and \(X_{2}\) and variances \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\). If \(X_{c}\) is the combined mean and \(\sigma_{c}^{2}\) is the combined variance of the sum of \(n_{1}\) and \(n_{2}\) observations, then the combined variance is calculated as follows:

\(\sigma_{c}^{2}=\frac{n_{1} \sigma_{1}^{2}+n_{2} \sigma_{2}^{2}+n_{1}\left(\overline{X_{1}}-\overline{X_{c}}\right)^{2}+n_{2}\left(\overline{X_{2}}-\overline{X_{c}}\right)^{2}}{n_{1}+n_{2}}\)

It can also be written as

\(\sigma_{c}^{2}=\frac{n_{1}\left[\sigma_{1}^{2}+\left(\overline{X_{1}}-\overline{X_{c}}\right)^{2}\right]+n_{2}\left[\sigma_{2}^{2}+\left(\overline{X_{2}}-\overline{X_{c}}\right)^{2}\right]}{n_{1}+n_{2}}\)

Standard Deviation

The standard deviation measures the spread of statistical data. The method of determining the deviation of data points is used to calculate the degree of dispersion. The symbol \(\sigma\) is used to represent it. Standard deviation is the square root of variance.

Both measurements have a wide distribution, but their units differ: The variance is expressed in squared units, whereas the standard deviation is expressed in the same units as the original values.

Combined Standard Deviation

Same as standard deviation, the combined standard deviation is the square root of the combined variance. Assume we have two sets of data of \(n_{1}\) and \(n_{2}\) observations, respectively, with averages \(X_{1}\) and \(X_{2}\) and standard deviations \(S_{1}^{2}\) and \(S_{2}^{2}\). If \(X_{c}\) is the combined mean and \(S_{c}^{2}\) is the combined standard deviation of the sum of \(n_{1}\) and \(n_{2}\) observations, then the combined standard deviation is calculated as follows:

\(S_{c}=\sqrt{\frac{n_{1}\left[S_{1}^{2}+\left(\overline{X_{c}}-\overline{X_{1}}\right)^{2}\right]+n_{2}\left[S_{2}^{2}+\left(\overline{X_{c}}-\overline{X_{2}}\right)^{2}\right]}{n_{1}+n_{2}}}\)

This can also be written as

\(S_{c}=\sqrt{\frac{n_{1}\left[S_{1}^{2}+d_{1}^{2}\right]+n_{2}\left[S_{2}^{2}+d_{2}^{2}\right]}{n_{1}+n_{2}}}\)

Here, \(d_{1}=\overline{X_{c}}-\overline{X_{1}}\) and \(d_{2}=\overline{X_{c}}-\overline{X_{2}}\)

Steps to Calculate Combined Standard Deviation

The first and foremost step in calculating the combined standard deviation is calculating the combined mean of the given data.

Step 1: Observe the number of observations in each group.

Consider the values of \(n_{1}, n_{2}, n_{3}, \ldots \ldots n_{i}\)

Step 2: Find the mean or average of each data group.

Mean \(=\frac{\text { Sum of values of observations }}{\text { Number of observations }}\)

Step 3: Find the combined mean of the data by using the formula.

\(x_{c}=\frac{n_{1} \overline{x_{1}}+n_{2} \cdot \overline{x_{2}}+n_{3} \bar{x}+\cdots}{n_{1}+n_{2}+n_{3}+\cdots}\)

Step 4: Find the variance of each data group.

\(\sigma^{2}=\frac{\sum\left(x_{i}-\bar{x}\right)^{2}}{n-1}\)

Step 5: Find the combined variance of the given data by using the formula.

\(\sigma_{c}^{2}=\frac{n_{1}\left[\sigma_{1}^{2}+\left(\overline{X_{1}}-\overline{X_{C}}\right)^{2}\right]+n_{2}\left[\sigma_{2}^{2}+\left(\overline{X_{2}}-\overline{X_{C}}\right)^{2}\right]+\cdots \cdots_{0}}{n_{1}+n_{2}+\cdots \cdots}\)

Step 6: Square root of the combined variance gives the combined standard deviation.

\(\sigma_{c}=\sqrt{\sigma_{c}^{2}}\)

Probability Distribution

The probability distribution determines the possible outcomes of any random event. It is also defined as the set of possible outcomes of any random experiment based on the underlying sample space. These parameters could be a set of real integers, vectors, or any other entities.

There are two main types of probability distributions utilised for diverse reasons and data creation processes.

• Probability Distributions: Normal or Cumulative
• Discrete or Binomial Probability Distribution

Coefficient of Variation

A type of relative measure of dispersion is the coefficient of variation. It is calculated by dividing the standard deviation by the mean. The coefficient of variation is a dimensionless quantity and is commonly expressed as a percentage. It aids in comparing two data sets based on the degree of variation.

The coefficient of variation can be calculated for both a sample and a population.

• Population Coefficient of Variation \(=\sigma_{\mu} \times 100\).
• Sample Coefficient of Variation \(=s_{\mu} \times 100\)

Where,

\(\sigma=\sqrt{\frac{\sum\left(x_{i}-\mu\right)^{2}}{N}}\) is the standard deviation of the population.

\(\mathrm{s}=\sqrt{\frac{\Sigma\left(x_{i}-\mu\right)^{2}}{N-1}}\) is the standard deviation of the sample.

Solved Examples

Check below the solved examples on Mean of Combined Distributions and Combined Standard Deviation:

Q.1. The following results were obtained from an analysis of monthly wages paid to employees in two service businesses, \(X\) and \(Y\) :

Which company has the most variation in individual pay among all wage earners combined?
Sol:
Here, we need to calculate the combined variance.
To calculate the combined variance, first we need to find out the combined mean.
Combined mean \(=x_{c}=x_{12}=\frac{n_{1}, \overline{x_{1}}+n_{2} \overline{x_{2}}}{n_{1}+n_{2}}\)
\(x_{c}=\frac{550 \times 5000+650 \times 4500}{550+650}\)
\(x_{c}=\frac{2750000+29250000}{1200}\)
\(\therefore x_{c}=Rs .4729 .166\)
Combined variance, \(S_{c}^{2}=\frac{n_{1}\left[S_{1}^{2}+\left(\overline{X_{1}}-\overline{X_{c}}\right)^{2}\right]+n_{2}\left[S_{2}^{2}+\left(\overline{X_{2}}-\overline{X_{c}}\right)^{2}\right]}{n_{1}+n_{2}}\)
\(=\frac{550\left(900+(4729.2-5000)^{2}\right)+650\left(1600+(4729.2-4500)^{2}\right)}{(550+650)}\)
\(\therefore S_{c}^{2}=633445\)
Hence, the combined variance is Rs. \(6,33,445\)

Q.2. In class \(X\), there are three sections \(A, B\), and \(C\), each with \(25,40\) , and \(35\) students. Sections \(A, B\), and \(C\) received \(70\) percent, \(65\) percent, and \(50\) percent of the total scores, respectively. Calculate the average grade in class \(X\).
Sol:
From the given data,
\(n_{1}=25, n_{2}=40\) and \(n_{3}=35\)
\(x_{1}=70 \%, x_{2}=65 \%\) and \(x_{3}=50 \%\)
The combined mean or average is given by
\(x_{123}=\frac{n_{1} \cdot \overline{x_{1}}+n_{2} \cdot \overline{x_{2}}+n_{3} \overline{x_{3}}}{n_{1}+n_{2}+n_{3}}\)
\(=\frac{25 \times 70+40 \times 65+35 \times 50}{25+40+35}\)
\(=\frac{1750+2600+1750}{100}\)
\(=\frac{6100}{100}\)
\(\therefore x_{123}=61 \%\)
Hence, the average mark of the class is \(61 \%\).

Q.3. The mean of the monthly salary of employees in a company was \(Rs 600\). The means of the monthly salaries paid to male and female employees were \(Rs .620\) and \(Rs .520\), respectively. Find the percentage of male to female employees.
Sol:
Let \(n_{1}-\) male employees and \(n_{2}-\) female employees combined Mean \(=600\)
\(\overline{X_{1}}=620, \overline{X_{2}}=520\) and \(X_{c}=600\)
Using the formula:
\(x_{c}=x_{12}=\frac{n_{1} \cdot \overline{x_{1}}+n_{2} \overline{x_{2}}}{n_{1}+n_{2}}\)
\(600=\frac{620 n_{1}+520 n_{2}}{n_{1}+n_{2}}\)
\(600\left(n_{1}+n_{2}\right)=620 n_{1}+520 n_{2}\)
\(600 n_{1}+600 n_{2}=620 n_{1}+520 n_{2}\)
\(600 n_{2}-520 n_{2}=620 n_{1}-600 n_{1}\)
\(80 n_{2}=20 n_{1}\)
\(\frac{n_{2}}{n_{1}}=\frac{20}{80}=\frac{1}{4}\)
\(\frac{n_{1}}{n_{2}}=\frac{4}{1}\)
\(\therefore n_{1}: n_{2}=4: 1\)
Percentage of male employees \(=\frac{4}{4+1} \times 100=80 \%\)
Percentage of female employees \(=\frac{1}{4+1} \times 100=20 \%\)
Hence, the percentage of male and female employees is \(80 \%\) and \(20 \%\), respectively.

Q.4. The mean and variance of daily salaries for a group of \(50\) male workers are \(63\) dollars and \(9\) dollars, respectively. These figures are \(54\) dollars and \(6\) dollars for a group of \(40\) female workers. Calculate the variance of the group of \(90\) persons.
Sol:
Given:
\(n_{1}=50, \overline{X_{1}}=63, S_{1}^{2}=9^{2}=81\)
\(n_{2}=40, \overline{X_{2}}=54, S_{2}^{2}=6^{2}=36\)
In order to calculate the combined variance, first we need to find out the combined mean.
Combined mean \(=x_{c}=x_{12}=\frac{n_{1} \cdot \overline{x_{1}}+n_{2} \cdot \overline{x_{2}}}{n_{1}+n_{2}}\)
\(x_{c}=\frac{50 \times 63+40 \times 54}{50+40}\)
\(x_{c}=\frac{3150+2160}{90}\)
\(\therefore x_{c}=59\) dollars
Combined variance, \(S_{c}^{2}=\frac{n_{1}\left[S_{1}^{2}+\left(\overline{X_{1}}-\overline{X_{c}}\right)^{2}\right]+n_{2}\left[S_{2}^{2}+\left(\overline{X_{2}}-\overline{X_{c}}\right)^{2}\right]}{n_{1}+n_{2}}\)
\(=\frac{50\left(81+(59-63)^{2}\right)+40\left(36+(59-54)^{2}\right)}{(50+40)}\)
\(\therefore S_{c}^{2}=81\)
Hence, the combined variance is \(81\) dollars.

Q.5. The mean and standard deviation of two distributions of \(100\) and \(150\) things are \(50,5\) and \(40,6\) correspondingly. Calculate the standard deviation of all \(250\) items combined.
Sol:
From the question,
\(n_{1}=100, \overline{X_{1}}=50, \sigma_{1}^{2}=5^{2}=25\)
\(n_{2}=150, \overline{X_{2}}=40, \sigma_{2}^{2}=6^{2}=36\)
To calculate the combined standard deviation, first, we need to find out the combined mean.
Combined mean \(=x_{c}=x_{12}=\frac{n_{1}, \overline{x_{1}}+n_{2} \overline{x_{2}}}{n_{1}+n_{2}}\)
\(=\frac{100 \times 50+150 \times 40}{100+150}\)
\(=\frac{5000+6000}{250}\)
\(=44\)
Combined standard deviation, \(\sigma_{c}=\sqrt{\frac{n_{1}\left[\sigma_{1}^{2}+\left(\overline{X_{c}}-\overline{X_{1}}\right)^{2}\right]+n_{2}\left[\sigma_{2}^{2}+\left(\overline{X_{c}}-\overline{X_{2}}\right)^{2}\right]}{n_{1}+n_{2}}}\)
\(=\sqrt{\frac{100\left(25+(44-50)^{2}\right)+150\left(36+(44-40)^{2}\right)}{50+40}}\)
\(=\sqrt{\frac{100 \times 61+150 \times 52}{250}}\)
\(=\sqrt{\frac{13900}{250}}\)
\(\therefore \sigma_{c}=7.456\)

Summary

The mean of the data is the ratio of the sum of observations to the number of observations. The combined mean of two or more data groups can be found by using the formula. For this, we must know the number of observations and the mean of each data group.

Variance and standard deviation are two statistical tools that measure the distribution of data about its mean. Combined variance is the variances of data groups, while combined standard deviation is the square root of combined variance.

The probability distribution is the study of all possible outcomes of a particular event. The coefficient of variation is the ratio of the standard deviation to the mean of the data.

Frequently Asked Questions (FAQs)

Below are the frequently asked questions on Mean of Combined Distributions and Combined Standard Deviation:

Q.2. How do you calculate combined mean and combined standard deviation?
Ans: Combined mean \(= \frac{{{n_1}\overline {{x_1}}  + {n_2}\overline {{x_2}}  + { \ldots _{ \ldots  \ldots  \ldots  + }} + {n_i}{{\bar x}_l}}}{{{n_1} + {n_2} + { \cdots _{ \ldots  \ldots  \ldots }} + {n_i}}}\)
Combined standard deviation \(\left(S_{c}\right)=\sqrt{\frac{n_{1}\left[\sigma_{1}^{2}+\left(\overline{X_{c}}-\overline{X_{1}}\right)^{2}\right]+n_{2}\left[\sigma_{2}^{2}+\left(\overline{X_{c}}-\overline{X_{2}}\right)^{2}\right]+\cdots \ldots . . n_{i}\left[\sigma_{i}^{2}+\left(\overline{X_{c}}-\overline{X_{1}}\right)^{2}\right]}{n_{1}+n_{2}+\ldots \ldots \ldots+n_{i}}}\)

Q.3. What is the relation between the variance and standard deviation?
Ans: Standard deviation is the square root of the variance.

Q.4. How do you calculate the combined variance?
Ans: Combined variance is calculated using the formula for the two data groups
\(\sigma_{c}^{2}=\frac{n_{1}\left[S_{1}^{2}+\left(\overline{X_{1}}-\overline{X_{c}}\right)^{2}\right]+n_{2}\left[S_{2}^{2}+\left(\overline{X_{2}}-\overline{X_{c}}\right)^{2}\right]}{n_{1}+n_{2}}\)

Q.5. What is the coefficient of variation?
Ans: The coefficient of variation is the ratio of the standard deviation to the mean. It is expressed as the percentage.

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