Let the mean and the standard deviation of the observations 2, 3, 3, 4, 5, 7, a, b be 4 and √2 respectively. Then the mean deviation about the mode of these observations is:

Step 1: Calculate the sum of observations and sum of squares

First, we calculate the sum of all observations and the sum of the squares of all observations. This will help us use the given mean and standard deviation formulas to find the values of $a$ and $b$. There are $n=8$ observations in total.

Step 2: Use the mean formula to find a relationship between a and b

The mean of the observations is given as 4. We use the formula for the mean, $\bar{x} = \frac{\sum x_i}{n}$, and substitute the sum of observations and the number of observations to find the first equation relating $a$ and $b$.

Step 3: Use the standard deviation formula to find a and b

The standard deviation is given as $\sqrt{2}$. We use the formula for variance, $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$, to set up a second equation. By solving the system of equations $a+b=8$ and $a^2+b^2=32$, we find that $a=4$ and $b=4$.

Step 4: Identify the mode of the observations

Now that we have found $a=4$ and $b=4$, the complete set of observations is $2, 3, 3, 4, 4, 4, 5, 7$. The mode is the observation that appears most frequently. In this set, the number 4 appears three times, which is more than any other number.

Step 5: Calculate the mean deviation about the mode

The mode is 4. The mean deviation about the mode is calculated by finding the absolute difference between each observation and the mode, summing these differences, and then dividing by the total number of observations. This gives us a mean deviation of 1.

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.

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