Probability theory

1. Lecture 3
Topics: Outlier in Box Plot
Mean, Median and Spread data
SADAF SALEEM
Department of Computer Sciences
GIFT University, Pakistan
Course Title: Probability Theory
Course Code: MATH-313
Program: BS MATH
Semester: Spring 2023

2. Box-and-Whisker Plot
Outlier:
• Sometime in box and whisker plot we mark another point called outlier.
• In a box plot, an asterisk (*) identifies an outlier.
• An outlier is a value that is inconsistent with the rest of the data. It is
defined as a value that is more than 1.5 times the interquartile range
smaller than Q1 or larger than Q3.

3. Box-and-Whisker Plot
Example 1:
10.2, 14.1, 14.4. 14.4, 14.4, 14.5, 14.5, 14.6, 14.7, 14.7, 14.7, 14.9, 15.1, 15.9,
16.4
Mark any outlier if exist.
Solution:
Step 1: arrange data
Step 2: Find Q2, Q1, Q3
• Q2 = 14.6
• Q1 = 14.4
• Q3 = 14.9
Step 3: Find IQR (inter quartile range)
IQR = 14.9 – 14.4 = 0.5
Step 4: Calculated Lower limit and Upper limit
• Q1 – 1.5 ×IQR = 14.4 – 0.75 = 13.65
• Q3 + 1.5×IQR = 14.9 + 0.75 = 15.65.
Step 5: Mark outliers
Then the outliers are at:
10.2, 15.9, and 16.4

4. Box-and-Whisker Plot
Example 1:
21, 23, 24, 25, 29, 33, 49
Mark any outlier if exist.
Solution:
Step 1: arrange data
Step 2: Find Q2, Q1, Q3
• Q2 = 25
• Q1 = 23;
• Q3 = 33
Step 3: Find IQR (inter quartile range)
IQR = 33 – 23 = 10
Step 4: Calculated Lower limit and Upper limit
• Q1 – 1.5 ×IQR = 23 – 1.5×10 = 23 – 15 = 8
• Q3 + 1.5×IQR = 33 + 1.5×10 = 33 + 15 = 48
Step 5: Mark outliers

5. Box-and-Whisker Plot
Example 3:
Let the data range be 199, 201, 236, 269,271,278,283,291, 301, 303, and 341.
Mark any outlier if exist.
Solution:
• Q2 = 278
• Q1 = 236
• Q3 = 301
Step 3: Find IQR (inter quartile range)
IQR = 301-236=65
Step 4: Calculated Lower limit and Upper limit
• Q1 – 1.5 ×IQR = 138.5
• Q3 + 1.5×IQR =398.5
No outlier exist

6. Box-and-Whisker Plot
Example 3:
The box plot below shows the amount spent for books and supplies per year
by students at four-year public colleges.
a. Estimate the median amount spent.
b. Estimate the first and third quartiles for the amount spent.
c. Estimate the interquartile range for the amount spent.
d. Beyond what point is a value considered an outlier?
e. Identify any outliers and estimate their value.

7. Box-and-Whisker Plot
Solution:

8. Analysis of Data

9. Sample Mean
• Average value of the sample
• Influenced by extreme values
• Represented by 𝑥
• Formula:
• Example 1:
• 1.7, 2.2, 3.9, 3.11, 14.7
• 𝒙 =
1.7+2.2+3.9+3.11+14.7
5
• 𝒙 = 5.12

10. Sample Mean: Class Practice

11. Sample Median
• Central tendency of the Sample
• Don't influenced by extreme values
• Represented by 𝑥
• Formula:
• Arrange observation in increasing order
• Example 1: 1.7, 2.2, 3.9, 3.11, 14.7
• 1.7, 2.2, 3.11, 3.90, 14.7
• 𝒙 = 𝟑. 𝟏𝟏

12. Sample Median: Class Practice

13. SADAF SALEEM
Department of Computer Sciences
GIFT University, Pakistan