Find the mean deviation about the median for the following data: $x_{i}$: 3, 6, 9, 12, 13, 15, 21, 22 $f_{i}$: 3, 4, 5, 2, 4, 5, 4, 3.

Step 1: Calculate the cumulative frequency and total number of observations

First, we need to organize the given data in a frequency distribution table and calculate the cumulative frequency ($cf$). The cumulative frequency for each class is the sum of its frequency and the frequencies of all preceding classes. We also find the total number of observations, $N$, by summing all frequencies.

Step 2: Determine the median

For discrete data, the median is the value of the $(\frac{N}{2})^{\text{th}}$ observation if $N$ is even. Here, $N=30$, so we look for the $15^{\text{th}}$ observation. By checking the cumulative frequency column, we find that the $15^{\text{th}}$ observation falls in the class where $x_i = 13$.

Step 3: Calculate absolute deviations from the median

Next, we calculate the absolute deviation of each observation from the median, $|x_i - M|$, where $M=13$. Then, we multiply each absolute deviation by its corresponding frequency, $f_i|x_i - M|$, and sum these products to get $\sum f_i|x_i - M|$.

Step 4: Calculate the mean deviation about the median

Finally, we calculate the mean deviation about the median using the formula: $\frac{\sum f_i|x_i - M|}{N}$. We substitute the sum of the products of frequencies and absolute deviations, which is 149, and the total number of observations, which is 30.

Step 5: Simplify the result

Dividing 149 by 30 gives us the mean deviation about the median.

More from this chapter

• Calculate the mean and variance of the first n natural numbers.
• For a statistical data x_1, x_2,, x_10 of 10 values, a student obtained the mean as 5.5 and Σ_i=1^(10) x_i² = 371. He later found that he had noted two values i
• The mean and standard deviation of 100 observations were calculated as 40 and 5.1, respectively by a student who took by mistake 50 instead of 40 for one observ
• Let the mean and variance of five observations x_1=1, x_2=3, x_3=a, x_4=7 and x_5=b, with a>b, be 5 and 10 respectively. Then the variance of the observations n
• The mean of 5 observations is 4.4 and their variance is 8.24. If three of the observations are 1, 2 and 6, find the other two observations.
• If each of the observation x_1, x_2,..., x_n is increased by ' a ', where a is a negative or positive number, show that the variance remains unchanged.
• The variance of the numbers 8, 21, 34, 47,, 320 is:
• The variance of 20 observations is 5. If each observation is multiplied by 2, find the new variance of the resulting observations.

All Statistics solutions →