When the population distribution is normal, the statistic median {|X_1 - X|, ...

Question

When the population distribution is normal, the statistic median {|X_1 - X|, ..., |X_n - X|}/0.6745 can be used to estimate sigma. This estimator is more resistant to the effects of outliers (observations far from the bulk of the data) than is the sample standard deviation. Compute both the corresponding point estimate and s for the data below. 24.87 25.73 26.26 26.43 26.67 27.16 27.32 27.55 27.75 27.85 27.97 28.05 28.29 28.50 28.51 28.88 29.12 29.14 29.32 30.83 point estimate sample standard deviation s =

Explanation / Answer

SOl:

Arrange the data in Ascending order.

Middle most value is median

'which is the best point estimate

The median of the data set is 27.91.

Explanation

The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

24.87   25.73   26.26   26.43   26.67   27.16   27.32   27.55   27.75   27.85   27.97   28.05   28.29   28.50   28.51   28.88   29.12   29.14   29.32   30.83   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Median=27.85+27.972=27.91

point estimate=27.91

Sample standard deviation=1.392

Explanation

To find standard deviation we use the following formula

=(ximedian)2n1

We will compute this formula in 4 steps.

Step 1: Find the median.

Here median=27.91.

Step 2: Create the following table.

Step 3: Find the sum of numbers in the last column to get.

(xiX¯¯¯)2=36.822

Step 4: Calculate s using the above formula.

s=sqrt[(xiX¯¯¯)2/n1

=sqrt[36.822/(201)]

=1.392

s=1.392(rounding to 3 decimals)

data median data-median (data-median)^2 24.87 27.91 -3.04 9.2416 25.73 27.91 -2.18 4.7524 26.26 27.91 -1.65 2.7225 26.43 27.91 -1.48 2.1904 26.67 27.91 -1.24 1.5376 27.16 27.91 -0.75 0.5625 27.32 27.91 -0.59 0.3481 27.55 27.91 -0.36 0.1296 27.75 27.91 -0.16 0.0256 27.85 27.91 -0.06 0.0036 27.97 27.91 0.06 0.0036 28.05 27.91 0.14 0.0196 28.29 27.91 0.38 0.1444 28.5 27.91 0.59 0.3481 28.51 27.91 0.6 0.36 28.88 27.91 0.97 0.9409 29.12 27.91 1.21 1.4641 29.14 27.91 1.23 1.5129 29.32 27.91 1.41 1.9881 30.83 27.91 2.92 8.5264 36.822

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