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Median formula class 10th

Class 10
Median formula class 10th

Median Formula Class 10th is included in statistics, to measure of central tendency that is often used to describe the typical value of a set of observations.. The median divides the observations into two halves: half of the observations are smaller than the median, and half are larger.

The formula for calculating the median depends on whether the number of observations in the set is odd or even. In the case of an odd number of observations, the median is the middle value of the ordered set. In the case of an even number of observations, the median is the average of the two middle values of the ordered set.

The median is a useful measure of central tendency because it is not affected by extreme values or outliers in the data set. It is commonly used in fields such as economics, finance, and healthcare to describe the typical value of a set of observations.

Steps to calculate median (Grouped data)

Calculating the median for grouped data involves the following steps:

  1. Arrange the set of observations in ascending or descending order.
  2. Determine the number of observations, denoted by $n$.
  3. If $n$ is odd, the median is the middle observation. If $n$ is even, the median is the average of the two middle observations.
  4. To find the middle observation(s), divide $n$ by 2 to get the position of the middle observation(s) in the ordered set. If $n$ is odd, the middle observation is at position $\frac{n+1}{2}$. If $n$ is even, the two middle observations are at positions $\frac{n}{2}$ and $\frac{n}{2}+1$.
  5. If $n$ is odd, the median is the observation at the middle position. If $n$ is even, the median is the average of the two observations at the middle positions.

For example, suppose we have the following set of observations: 7, 2, 3, 8, 5. To calculate the median, we follow these steps:

  1. Arrange the observations in ascending order: 2, 3, 5, 7, 8.
  2. Determine the number of observations, $n=5$.
  3. Since $n$ is odd, the median is the middle observation.
  4. The middle observation is at position $\frac{n+1}{2}=\frac{5+1}{2}=3$.
  5. The median is the observation at the middle position, which is 5.

Therefore, the median of the set of observations 7, 2, 3, 8, 5 is 5.

Examples on Median

Example 1: Find the median of the following set of observations: 10, 15, 20, 25, 30, 35.

Solution:

  1. Arrange the observations in ascending order: 10, 15, 20, 25, 30, 35.
  2. Determine the number of observations, $n=6$.
  3. Since $n$ is even, the median is the average of the two middle observations.
  4. The two middle observations are at positions $\frac{n}{2}=\frac{6}{2}=3$ and $\frac{n}{2}+1=\frac{6}{2}+1=4$.
  5. The median is the average of the two observations at the middle positions: $\frac{20+25}{2}=22.5$.

Therefore, the median of the set of observations 10, 15, 20, 25, 30, 35 is 22.5.

Example 2 : Find the median of the following set of observations: 2, 2, 3, 3, 3, 4, 5, 6.

Solution:

  1. Arrange the observations in ascending order: 2, 2, 3, 3, 3, 4, 5, 6.
  2. Determine the number of observations, $n=8$.
  3. Since $n$ is even, the median is the average of the two middle observations.
  4. The two middle observations are at positions $\frac{n}{2}=\frac{8}{2}=4$ and $\frac{n}{2}+1=\frac{8}{2}+1=5$.
  5. The median is the average of the two observations at the middle positions: $\frac{3+3}{2}=3$.

Therefore, the median of the set of observations 2, 2, 3, 3, 3, 4, 5, 6 is 3.

Steps to calculate median (Ungrouped data)

The formula to calculate the median of ungrouped data is as follows:

  1. Arrange the observations in ascending or descending order.
  2. Determine the number of observations, denoted by $n$.
  3. If $n$ is odd, the median is the middle observation. If $n$ is even, the median is the average of the two middle observations.

If $X_{(1)}, X_{(2)}, ..., X_{(n)}$ represent the ordered observations, then the median can be calculated as:

  • If $n$ is odd: Median = $X_{\left(\frac{n+1}{2}\right)}$
  • If $n$ is even: Median = $\frac{X_{\left(\frac{n}{2}\right)} + X_{\left(\frac{n}{2}+1\right)}}{2}$

Here, $X_{(k)}$ represents the $k$th observation in the ordered set.

For example, consider the following set of ungrouped data: 8, 5, 6, 4, 2, 7, 3, 1.

  1. Arrange the observations in ascending order: 1, 2, 3, 4, 5, 6, 7, 8.
  2. Determine the number of observations, $n=8$.
  3. Since $n$ is even, the median is the average of the two middle observations.
  4. The two middle observations are at positions $\frac{n}{2}=\frac{8}{2}=4$ and $\frac{n}{2}+1=\frac{8}{2}+1=5$.
  5. The median is the average of the two observations at the middle positions: $\frac{4+5}{2}=4.5$.

Therefore, the median of the set of ungrouped data 8, 5, 6, 4, 2, 7, 3, 1 is 4.5.

Example 1: Find the median of the following set of observations: 5, 8, 3, 9, 2.

Solution:

  1. Arrange the observations in ascending order: 2, 3, 5, 8, 9.
  2. Determine the number of observations, $n=5$.
  3. Since $n$ is odd, the median is the middle observation.
  4. The middle observation is at position $\frac{n+1}{2}=\frac{5+1}{2}=3$.
  5. The median is the observation at the middle position, which is 5.

Therefore, the median of the set of observations 5, 8, 3, 9, 2 is 5.

Example 2: Find the median of the following set of observations: 10, 15, 20, 25, 30, 35.

Solution:

  1. Arrange the observations in ascending order: 10, 15, 20, 25, 30, 35.
  2. Determine the number of observations, $n=6$.
  3. Since $n$ is even, the median is the average of the two middle observations.
  4. The two middle observations are at positions $\frac{n}{2}=\frac{6}{2}=3$ and $\frac{n}{2}+1=\frac{6}{2}+1=4$.
  5. The median is the average of the two observations at the middle positions: $\frac{20+25}{2}=22.5$.

Therefore, the median of the set of observations 10, 15, 20, 25, 30, 35 is 22.5.

Example 3: Find the median of the following set of observations: 2, 2, 3, 3, 3, 4, 5, 6.

Solution:

  1. Arrange the observations in ascending order: 2, 2, 3, 3, 3, 4, 5, 6.
  2. Determine the number of observations, $n=8$.
  3. Since $n$ is even, the median is the average of the two middle observations.
  4. The two middle observations are at positions $\frac{n}{2}=\frac{8}{2}=4$ and $\frac{n}{2}+1=\frac{8}{2}+1=5$.
  5. The median is the average of the two observations at the middle positions: $\frac{3+3}{2}=3$.

Therefore, the median of the set of observations 2, 2, 3, 3, 3, 4, 5, 6 is 3.

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