Question

Identify potential outliers, if any, for the given data. The test scores of 15 students are listed below. 38 41 56 65 66 68 70 72 74 77 78 82 87 90 99 Steps: 1. Find Q1, Q3, and IQR 2. Find Q1 - 1.5 x IQR 3. Find Q3 +1.5 x IQR 4. Result from step 2 < Possible allowed data <Result from step 3 5. Compare step 4 to your data, find the outliers if they exist 38 38,99 None 38,41

Answer 1

Step 1)

Let's find median for given data first.

Median = ordered value , if n odd

th 1 2+1 2

= Average of $_{\left ( \frac{n}{2} \right )^{th}}$ and $_{\left ( \frac{n}{2} +1\right )^{th}}$ ordered value , if n even.  

Given data : 38,41,56,65,66,68,70,72,74,77,78,82,87,90,99  

n = 15 ---> odd.

So, Median = ordered value

th 1 2+1 2

Median = $_{\left ( \frac{15+1}{2} \right )^{th}}$ ordered value ,

Median = 8th ordered value

Median = 72

Now let's find first Quartile (Q1)

Q1 is the median of lower half.

Lower half is the all data below median.

Lower half : 38,41,56,65,66,68,70, n = 7

So,Q1= ordered value

th 1 2+1 2

Q1 = $_{\left ( \frac{7+1}{2} \right )^{th}}$ ordered value

Q1 = 4th ordered value.

Q₁ = 65

Now let's find third Quartile (Q3)

Q3 is the median of upper half.

Upper half is the all data above median.

Upper half : 74,77,78,82,87,90,99

Q3 = 82

IQR = 82- 65 = 17

Step 2)

$Q_1 - 1.5\times IQR$

$65 - 1.5\times 17$

Step 3)

$Q_3 + 1.5\times IQR$

$82 + 1.5\times 17$

Step 4)

< ( 41,56,65,66,68,70,72,74,77,78,82,87,90,99 ) <

Outlier is the value outside the interval ( 39.5 , 107.5 )

So, Outlier = 38