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:
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Step-by-Step Solution
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, xˉ=n∑xi, 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 2. We use the formula for variance, σ2=n∑xi2−(xˉ)2, to set up a second equation. By solving the system of equations a+b=8 and a2+b2=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.