3. Percentile
• A percentile is a measure used in statistics indicating the
value below which a given percentage of observations in a
group of observations fall.
• The formula for percentile is:
4. • The 25th percentile is also called the first quartile.
The 50th percentile is generally the median (if you’re using the third
definition—see below).
The 75th percentile is also called the third quartile.
The difference between the third and first quartiles is
the interquartile range.
5. Example:
The scores for student are 40, 45, 49, 53, 61, 65, 71, 79, 85, 91.
What is the percentile for score 71?
Solution:
No. of. scores below 71 = 6
Total no. of. scores = 10
The formula for percentile is given as,
Percentile of 71
= 6/10 × 100
= 0.6 × 100
= 60
6. Quartiles
Quartiles are the values that divide a list of numbers into quarters:
• Put the list of numbers in order
• Then cut the list into four equal parts
• The Quartiles are at the "cuts"
7. Example: 5, 7, 4, 4, 6, 2, 8
• Put them in order: 2, 4, 4, 5, 6, 7, 8
• Cut the list into quarters:
And the result is:
• Quartile 1 (Q1) = 4
• Quartile 2 (Q2), which is also the Median, = 5
• Quartile 3 (Q3) = 7
8. Interquartile Range
• The "Interquartile Range" is from Q1 to Q3:
• To calculate it just subtract Quartile 1 from Quartile 3.
9. Example: 2, 4, 4, 5, 6, 7, 8
The Interquartile Range is:
Q3 − Q1 = 7 − 4 = 3
10. Box and Whisker Plot
• We can show all the important values in a "Box and Whisker Plot", like
this:
11. Example: Box and Whisker Plot and
Interquartile Range for
Q. 4, 17, 7, 14, 18, 12, 3, 16, 10, 4, 4, 11
Step 1:
• Put them in order:
3, 4, 4, 4, 7, 10, 11, 12, 14, 16, 17, 18
Step 2:
• Cut it into quarters:
3, 4, 4 | 4, 7, 10 | 11, 12, 14 | 16, 17, 18
12. • In this case all the quartiles are between numbers:
• Quartile 1 (Q1) = (4+4)/2 = 4
• Quartile 2 (Q2) = (10+11)/2 = 10.5
• Quartile 3 (Q3) = (14+16)/2 = 15
• Also:
• The Lowest Value is 3,
• The Highest Value is 18
13. • So now we have enough data for the Box and Whisker Plot:
• And the Interquartile Range is:
=> Q3 − Q1 = 15 − 4 = 11
14. Standard Score or Z-Score
• A standard score or z score is used when direct comparison of raw
scores is impossible.
• A standard score or z score for a value is obtained by subtracting the
mean from the value and dividing the result by the standard
deviation.
15. • Obtained by subtracting the mean from the value and dividing the
result by the standard deviation.
• The symbol for the standard score is z.
tan
value mean
z
s dard deviation
x x
z for samples
s
16. Example:
A national achievement test is administered annually to 3rd
graders. The test has a mean score of 100 and a standard deviation
of 15. If Jane's z-score is 1.20, what was her score on the test?
17. Solution
• z = (X - μ) / σ
• where z is the z-score, X is the value of the element, μ is the mean of
the population, and σ is the standard deviation.
• Solving for Jane's test score (X), we get
• X = ( z * σ) + 100 = ( 1.20 * 15) + 100 = 18 + 100 = 118
