Download 2.5 Measures of Position

Statistics
2.5 Measures of Position
LEQ: What do the quartiles of a box-and-whisker plot tell you about the data?
Procedure:
1. Quartiles:
a. Definition 1: The 3 ___________________, Q1, Q2, and Q3,
approximately divide an ordered data set into 4 equal parts:



About ¼ of the data falls on or below the 1st quartile Q1.
About ½ of the data falls on or below the 2nd quartile Q2.
About ¾ of the data falls on or below the 3rd quartile Q3.
b. Example 1: Finding the quartiles of a data set:
The test scores of the 15 employees enrolled in a CPR training course are
listed. Find the 1st, 2nd, and 3rd quartiles of the test scores.
13 9 18 15 14 21 7 10 11 20 5 18 37 16 17
c. Example 2: Using technology to find quartiles:
The tuition costs (in thousands of dollars) for 25 universities are listed.
Use a calculator to find the 1st, 2nd, and 3rd quartiles. What can you
conclude from the result?
20 26 28 25 31 14 23 15 12 26 29 24 31
19 31 17 15 17 20 31 32 16 21 22 28
d. Definition 2: The __________________________ of a data set is the
difference between the 3rd and 1st quartiles.
Interquartile range (IQR) = ___________
e. Example 3: Finding the interquartile range:
Find the interquartile range of the 15 test scores given in the example 1.
What can you conclude from the result?
f. Guidelines: Drawing a Box-and-Whisker Plot:

Find the 5-number summary of the data set.
o
o
o
o
o
The minimum entry.
The 1st quartile Q1.
The median Q2.
The 3rd quartile Q3.
The maximum entry.

Construct a horizontal scale that spans the range of the data.

Plot the 5 numbers above the horizontal scale.

Draw a box above the horizontal scale from Q1 to Q3 and draw a
vertical line at the box at Q2.

Draw whiskers from the box to the min and max entries.
g. Example 4: Drawing a box-and-whisker plot:
Sketch a box-and-whiskers plot that represents the 15 test scores given in
example 1. What can you conclude from the display?
2. Percentiles and Other Fractiles:
Fractiles
Quartiles
Deciles
Percentiles
Summary
Divide a data set into 4 equal parts.
Divide a data set into 10 equal parts
Divide a data set into 100 equal parts
Symbols
Q1, Q2, Q3
D1, D2, D3,…,D9
P1, P2, P3,….,P99
a. Example 5: Interpreting percentiles:
The ogive represents the cumulative frequency distribution for SAT test
scores of college-bound students in a recent year. What test score
represents the 64th percentile? How should you interpret this?
3. The Standard Score:
a. Definition 3: The ________________________, or _______________,
represents the number of standard deviations a given value x falls from
the mean. To find the z-score from a given value, use the following
formula.
b. Examples 6 & 7: Finding z-scores:
6. The mean speed of vehicles along a stretch of highway is 56 mph with
a standard deviation of 4 mph. You measure the speed of 3 cars traveling
along this stretch of highway as 62 mph, 47 mph, and 56 mph. Find the zscore that corresponds to each speed. What can you conclude?
7. The monthly utility bills in a city have a mean of $70 and a standard
deviation of $8. Find the z-scores that correspond to utility bills of $60,
$71, and $92. What can you conclude?
c. Example 8: Comparing z-scores from different data sets:
During the 2003 regular season the Kansas City Chiefs, one of 32 teams
in the National Football League (NFL), scored 63 touchdowns. During the
2003 regular season the Tampa Bay Storm, one of 16 teams in the Arena
Football League (AFL), scored 119 touchdowns. The mean number of
touchdowns in the NFL is 37.4, with a standard deviation of 9.3. The mean
number of touchdowns in the AFL is 111.7, with a standard deviation of
17.3. Find the z-score that corresponds to the number of touchdowns for
each team. Then compare your results.
4. HW: p. 101 (3 – 36 mo3)