This document provides information about measures of position such as quartiles, percentiles, and z-scores. It defines these terms and shows examples of how to calculate them. Quartiles divide a data set into 4 equal parts, with Q1 being the lower quartile, Q2 (the median) dividing the data set in half, and Q3 being the upper quartile. Percentiles such as P25, P50, and P75 indicate the values below which a certain percentage of observations in a data set fall. Z-scores measure a data point's position relative to the mean in terms of standard deviations and can be positive or negative. Box plots provide a visual representation of a data set using the minimum, quartiles,

1. Measures of Position
– Quartiles and
Percentiles

2. Objective
To be able to find Quartiles,
Percentiles, and Z-Scores.
To be able to draw box plots.

3. Relevance
To be able to evaluate our
relative position when
interested in comparing
performance and knowing a
ranking.

4. Position
 Used to describe the position of a data
value in relation to the rest of the data.
 Types:
1. Quartiles
2. Percentiles
3. Deciles

5. Quartiles……
 Values of the variables are divided into
quarters – 4 equal parts. They are called
Q1
Q2
Q3

6. Q1 – Lower Quartile
At most, 25% of data is smaller than Q1.
It divides the lower half of a data set in half.

7. Q2 - Median
 The median divides the data set in half.
 50% of the data values fall below the
median and 50% fall above.

8. Q3 – Upper Quartile
 At most, 25% of data is larger than Q3.
 It divides the upper half of the data set in
half.

9. Q1 Med Q3
25% 25% 25% 25%

10. Example……
 Find Q1, median, and Q3.
 3, 9, 12, 13, 15, 17, 19, 20, 24
 Put numbers in L1 – Run “Stat Calc One-
variable stats L1” – scroll all the way
down.

12. Percentiles……
 Values of the variable that divide a ranked
set into 100 subsets.
 For example, P30 would be at 30%.

13.  NOTE:
 Q1 = P25
 Q2 = P50 = Median
 Q3 = P75

14. Percentile Example……
 The 78th percentile means 78% are
smaller than the given value.
 Does making the 80th percentile mean that
you made an 80% on test?

15. Answer……
 No.
 The 80th percentile would mean that a
person did better than 80% of the students
who took the same test.

16. Value that corresponds to the kth
percentile……
This is a position locator formula.
C = np/100

17. If C is a Decimal…..
 If the value you find for “C” is NOT a whole
number, then round up to the next whole
number.
 This is the position where you will find your
answer.

18. If C is a whole number……
 If the value you find for “C” is a whole
number, then you must average “C” and
“C+1”.
 This gives you the position of the answer.

19. Example…..Use the following.
2,3,5,6,8,10,12,15,18,20
 Notice that the numbers are in ascending
order.
 Find P25 (the number in the 25th percentile
position.

20. 2,3,5,6,8,10,12,15,18,20
 Find P25……..
 C=np/100
 C = 10(25)/100 = 2.5
 Round up to the 3rd position.
 2,3,5,6,8,10,12,15,18,20
 The answer is 5.

21. Example 2 …..
2,3,5,6,8,10,12,15,18,20
 Find P60:
 C = np/100 = (10)(60)/100 = 6
 Average the numbers in the 6th and 7th
position together.
 2,3,5,6,8,10,12,15,18,20
 (10+12)/2 = 11
 The answer is 11.

22. Example 3…..
2,3,5,6,8,10,12,15,18,20
 Note: P75 is the same as Q3.
 C = (10)(75)/100= 7.5
 Round up to the 8th position.
 2,3,5,6,8,10,12,15,18,20
 The answer is 15.

23. Now you try……
 Use the following set of data to answer the
next 3 questions.
 3,4,4,6,8,10,10,12,12,12,13,15,15,15,16,
17,20,22,25,27

24.  A. Find P75.
 B. Find P30
 C. Find P23

25. Answer A….
 Find P75.
 C = (20)(75)/100 = 15
 The number will be the average of the 15th
and 16th position.
 3,4,4,6,8,10,10,12,12,12,13,15,15,15,16,
17,20,22,25,27
The answer is (16+17)/2 = 16.5

26. Answer B……
 Find P30.
 C = (20)(30)/100 = 6
 Average the 6th and 7th numbers.
 3,4,4,6,8,10,10,12,12,12,13,15,15,15,16,
17,20,22,25,27
 Answer: (10+10)/2 = 10

27. Answer C……
 Find P23.
 C = (20)(23)/100 = 4.6
 Round up to the 5th position.
 3,4,4,6,8,10,10,12,12,12,13,15,15,15,16,
17,20,22,25,27
 The answer is 8.

28. Midquartile……
 The number halfway between Q1 and Q3.
Midquartile = (Q1+Q3)/2

29. Example……
 If Q1 = 9 and Q3 = 16.5, find the
midquartile.
 Answer: (9+16.5)/2 = 25.5/2 = 12.75

30. Z-Scores

31. Z-Score……
 A z-score represents the number of
standard deviations a data value falls
above or below the mean.
 It is used as a way to measure relative
position.

32. Z-Score Formula……
deviation
st
mean
value
score
z
.
)
( 


s
x
x
z


.
2 places
decimal
to
scores
z
round
Please 

33. Can a z-score be negative?
 A positive z-score means that a score is
above the mean.
 A negative z-score means that a score is
below the mean.
 A z-score of 0 means that a score is the
exact same as the mean.
YES

34. Example……
 A student scored a 65 on a math test that
had a mean of 50 and a standard
deviation of 10. She scored 30 on a
history test with a mean of 25 and a
standard deviation of 5. Compare her
relative position on the two tests.

35. Answer……
 Math: z = (65-50)/10= 15/10 = 1.5
 History: z = (30-25)/5 = 5/5 = 1
 The student did better in math because
the z-score was higher.
deviation
st
mean
value
score
z
.




36. Example 2……
 Find the z-score for each test and state
which test is better.
 Test A:
 Test B:
5
40
38 

 s
x
x
10
100
94 

 s
x
x

37.  Test A: z = (38-40)/5 = -0.4
 Test B: z = (94-100)/10 = -0.6
 Which is higher?
 Test A is higher, therefore it is better.
 It has a higher relative position.

38. Example 3…….
 A sample has a mean of 200 and a
standard deviation of 25. Find the value of
x that corresponds to a z-score of 2.35.

39. Answer: A sample has a mean of 200 and a standard deviation of 25.
Find the value of x that corresponds to a z-score of 2.35.
 
75
.
258
75
.
58
200
)
25
)(
35
.
2
(
200
25
200
35
.
2









x
x
x
x
s
x
x
z

40. Box Plots
(Box and Whiskers)

41. How do we interpret a box
plot?
 It is a valuable source of easy-to-interpret
information about a sample of study.
 It can provide information about a
sample’s range, median, normality of the
distribution, and skew of the distribution.

42. Boxplot
 A graph of a set of data obtained by
drawing a horizontal line from the
minimum to maximum values with
quartiles labeled in between.
 It is a graphical plot of 5 specific values
called the 5-number summary.

43. 5-number summary…..
 1. minimum
 2. Q1
 3. median
 4. Q3
 5. maximum

44. Steps……
 1. Find the 5-number
summary…use your
calculator.
 2. Draw and label a
scale of equal
intervals.
 3. Place dots above
the 5 numbers
 4. Put a box around
Q1 and Q3.
 5. Draw a vertical line
through the median.
 6. Draw “whiskers”
from the minimum to
Q1 and maximum to
Q3.

45. Example……
 Draw a box plot of the following data.
33, 38, 43, 30, 29, 40, 51, 27, 42, 23, 31
Remember: You can put the numbers in
L1 of the calculator and run “stat calc
one-var stats L1” and then scroll down
to the bottom to get the 5-number
summary.

46. The 5-number summary…...
 Remember to
ALWAYS write down
on your paper exactly
what numbers you got
for the 5-number
summary.

47. Let’s draw the box plot by hand
together…..
 Min = 23
 Q1 = 29
 Median = 33
 Q3 = 42
 Max = 51

48. Example…..Now you try
 Draw a box plot for the following data.
3, 8, 15, 20, 22, 23, 23, 24, 28, 29, 29, 30,
35, 38

49. The 5-number summary…..

50. Let’s figure out the
numbers……

51. Skewness
 If the whisker to the right of the box is
longer than the one to the left in relation to
the median, there are more extreme
values towards the positive end and so the
distribution is positively skewed.
 Similarly, if the whisker to the left is longer,
the distribution is negatively skewed.

52. Skewed Left (Negative) or Right
(Positive)?
Right Skewed

53. With Histogram

54. Skewed Left (Negative) or Right
(Positive)?
Left Skewed

55. With
Histogram

56. Can we figure out the numbers for the
5-number summary?
 Answers:
 A: Min = 50
Q1 = 65
Med = 70
Q3 = 80
Max = 100
 B: Min = 40
Q1 = 60
Med = 70
Q3 = 85
Max = 100

57. Homework……
 Worksheet