Calculate The Mean Deviation Of The Data Given Below From The Median.$\[ \begin{tabular}{|c|c|c|c|c|c|} \hline CI & $10-20$ & $20-30$ & $30-40$ & $40-50$ & $50-60$ \\ \hline f & 5 & 4 & 5 & 4 & 2

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Introduction

In statistics, the mean deviation is a measure of the average distance between each data point and the median of the dataset. It is an important concept in understanding the spread or dispersion of data. In this article, we will calculate the mean deviation of the given data from the median.

Understanding the Data

The given data is presented in a frequency distribution table, which shows the class intervals (CI) and their corresponding frequencies (f). The data is divided into five class intervals: 10−2010-20, 20−3020-30, 30−4030-40, 40−5040-50, and 50−6050-60. The frequencies of each class interval are: 5, 4, 5, 4, and 2, respectively.

Calculating the Median

To calculate the mean deviation, we first need to find the median of the dataset. Since the data is presented in a frequency distribution table, we can use the following steps to find the median:

  1. Arrange the data in ascending order: First, we need to arrange the data in ascending order. However, since the data is presented in a frequency distribution table, we can use the mid-point of each class interval to represent the data.

  2. Calculate the cumulative frequency: Next, we need to calculate the cumulative frequency of each class interval. The cumulative frequency is the sum of the frequencies of all class intervals up to a certain point.

  3. Find the median class: The median class is the class interval that contains the median value. To find the median class, we need to find the cumulative frequency that is closest to the total frequency divided by 2.

  4. Calculate the median: Once we have found the median class, we can calculate the median value using the following formula:

    Median = L + \frac{\frac{n}{2} - C}{f} \times i

    where:

    • L is the lower limit of the median class
    • n is the total frequency
    • C is the cumulative frequency of the class preceding the median class
    • f is the frequency of the median class
    • i is the width of the median class

Calculating the Mean Deviation

Once we have found the median, we can calculate the mean deviation using the following steps:

  1. Calculate the absolute deviations: First, we need to calculate the absolute deviations of each data point from the median. The absolute deviation is the difference between each data point and the median.
  2. Calculate the mean of the absolute deviations: Next, we need to calculate the mean of the absolute deviations. The mean of the absolute deviations is the sum of the absolute deviations divided by the total frequency.

Step-by-Step Solution

Let's apply the above steps to the given data to calculate the mean deviation.

Step 1: Arrange the data in ascending order

CI f
10-20 5
20-30 4
30-40 5
40-50 4
50-60 2

Step 2: Calculate the cumulative frequency

CI f Cumulative Frequency
10-20 5 5
20-30 4 9
30-40 5 14
40-50 4 18
50-60 2 20

Step 3: Find the median class

The total frequency is 20. The cumulative frequency that is closest to 20/2 = 10 is 9. Therefore, the median class is 20-30.

Step 4: Calculate the median

The lower limit of the median class is 20. The frequency of the median class is 4. The width of the median class is 10. The cumulative frequency of the class preceding the median class is 5. The total frequency is 20.

Median = 20 + \frac{\frac{20}{2} - 5}{4} \times 10 Median = 20 + \frac{10 - 5}{4} \times 10 Median = 20 + \frac{5}{4} \times 10 Median = 20 + 12.5 Median = 32.5

Step 5: Calculate the absolute deviations

CI f Absolute Deviation
10-20 5 22.5
20-30 4 1.5
30-40 5 7.5
40-50 4 17.5
50-60 2 27.5

Step 6: Calculate the mean of the absolute deviations

The sum of the absolute deviations is 22.5 + 1.5 + 7.5 + 17.5 + 27.5 = 75. The total frequency is 20.

Mean of Absolute Deviations = \frac{75}{20} Mean of Absolute Deviations = 3.75

Conclusion

Q: What is the mean deviation?

A: The mean deviation is a measure of the average distance between each data point and the median of the dataset. It is an important concept in understanding the spread or dispersion of data.

Q: How do I calculate the mean deviation?

A: To calculate the mean deviation, you need to follow these steps:

  1. Arrange the data in ascending order: First, you need to arrange the data in ascending order. However, since the data is presented in a frequency distribution table, you can use the mid-point of each class interval to represent the data.

  2. Calculate the cumulative frequency: Next, you need to calculate the cumulative frequency of each class interval. The cumulative frequency is the sum of the frequencies of all class intervals up to a certain point.

  3. Find the median class: The median class is the class interval that contains the median value. To find the median class, you need to find the cumulative frequency that is closest to the total frequency divided by 2.

  4. Calculate the median: Once you have found the median class, you can calculate the median value using the following formula:

    Median = L + \frac{\frac{n}{2} - C}{f} \times i

    where:

    • L is the lower limit of the median class
    • n is the total frequency
    • C is the cumulative frequency of the class preceding the median class
    • f is the frequency of the median class
    • i is the width of the median class

  5. Calculate the absolute deviations: First, you need to calculate the absolute deviations of each data point from the median. The absolute deviation is the difference between each data point and the median.

  6. Calculate the mean of the absolute deviations: Next, you need to calculate the mean of the absolute deviations. The mean of the absolute deviations is the sum of the absolute deviations divided by the total frequency.

Q: What is the difference between the mean deviation and the standard deviation?

A: The mean deviation and the standard deviation are both measures of the spread or dispersion of data. However, they are calculated differently. The standard deviation is calculated using the formula:

Standard Deviation = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}

where:
-   x_i is each data point
-   \bar{x} is the mean of the data
-   n is the total frequency

The mean deviation, on the other hand, is calculated using the formula:

Mean Deviation = \frac{\sum |x_i - \bar{x}|}{n}

where:
-   x_i is each data point
-   \bar{x} is the median of the data
-   n is the total frequency

Q: When to use the mean deviation?

A: The mean deviation is used when the data is not normally distributed or when the data has outliers. It is also used when the data is presented in a frequency distribution table.

Q: What are the limitations of the mean deviation?

A: The mean deviation has several limitations. It is sensitive to outliers and can be affected by the presence of extreme values. It is also not a robust measure of dispersion and can be affected by the choice of the median.

Q: How to interpret the mean deviation?

A: The mean deviation can be interpreted as the average distance between each data point and the median of the dataset. It can be used to understand the spread or dispersion of data and to compare the spread of different datasets.

Conclusion

In this article, we have answered some frequently asked questions on calculating the mean deviation. The mean deviation is an important concept in understanding the spread or dispersion of data. It is used when the data is not normally distributed or when the data has outliers. However, it has several limitations and should be used with caution.