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Statistics
2.5: Measures of Position
Objective 1: I can find the quartiles of a data set, the interquartile range, and represent a
data set using a box-and-whisker plot.
In this section we will learn to use _____________ to specify the position of data in a data set.
____________ are numbers that divide a data set into ________ parts. Here are some common
fractiles that you may have heard of before.
Median:
Quartile:
Percentile:
Decile:
The three quartiles, ______, _______ (or the ___________) and ________, divide a data set into
____ equal parts. In this objective we are going to focus on finding the _______________
___________________ _______________________, which consists of the values:
Each section between two values contains ____________ of the data values.
TIY 1: Find the 5-number summary for the data set below.
19
38
22
7
18
24
29
15
22
27
25
16
11
12
31
9
20
28
First, Put the data in ORDER!!!
Using technology to find the 5-number summary:
STAT
CALC
1: 1-Vars Stats
Scroll down until you see Min, Q1, Md, Q3, and Max
31
14
TIY 2: The tuition costs (in thousands of dollars) for 25 universities are listed. Use a calculator
or computer to find the 5-number summary.
20
26
28
25
31
14
23
15
12
26
29
34
31
19
31
17
15
17
20
31
32
16
21
22
28
After finding the quartiles, you can then find the __________ __________________
___________ or ___________.
IQR =
TIY 3: Find the IQR for the data in TIY 1.
19
38
22
7
18
24
29
27
25
16
11
12
31
9
15
20
22
28
31
14
The IQR is a measure of variation that gives you an idea of how much the middle 50% of the
data varies. It can also be used to identify ______________. An outlier is any data value that
lies outside the interval found by
Formula:
*Are there any outliers in our data set from TIY 1?
One application of quartiles is to represent data with a __________________________ plot. To
draw a box-and-whisker plot,
1)
2)
3)
4)
5)
TIY 4: Create a box-and-whisker plot of the data in TIY 1.
19
38
22
7
18
24
29
15
22
27
25
16
11
12
31
9
20
28
31
14
Min:
Q1:
Md:
Q3:
Max:
Objective 2: I can interpret percentiles and other fractiles.
Besides using the median and quartiles to partition data, we can use other fractiles. They even
have their own notation.
Median
Deciles
____________ Parts
Quartiles
____________ Parts
Percentiles
____________ Parts
A fractile is used to compare data values in the same set. For example, if a child
places in the first quartile on a standardized test, then that child performed better
than ______ of the other children. When you receive your ACT scores back, they
grade you in each test category by _____________. So if you place in the 88 th
percentile for ACT math, then you performed better than _______ of other students
who took that math test.
TIY 5: The number of touchdowns scored by all Division 1A football teams are represented by
the ogive below. Use it to answer parts (a) and (b).
a) Find the percentile that corresponds to 40 touchdowns. Interpret this.
b) Find the number of touchdowns for a team that scored in the 75th percentile. Interpret this.
Objective 3: The standard score
When you know the mean and standard deviation of a data set, you can measure a data value’s
position in the data set with a _______________________ or ___________. The
_____________________, or z-score, represents the number of __________________________
a given value, x, falls from the _____________. The z-score for any x-value in a data set can be
found by the formula:
A _____________ z-score means the data value is __________________ the mean.
A _____________ z-score means the data value is __________________ the mean.
A z-score of _______ means the data value _____ the mean.
Z-scores are often used to compare data from different sets—such as ACT scores compared to
SAT scores. How can you tell which score is “better” for its data set?
Quick Example: Bob scored 27 on the ACT which has a mean of 21 and a SD of 3.5. He also
took the SAT math and scored 680, which has a mean of 500 and a SD of 100. On which test did
Bob perform better? (Always round z-score to 2 decimal places.)
Z-scores can also help us determine if a data value is an ______________. Recall that an outlier
is any data value that is __________________________________________________________.
TIY 6: 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. Would any of the bills be
considered outliers?
TIY 7: Alan Arkin won a Best Supporting Actor Oscar at age 72, while Jennifer Hudson won a
Best Supporting Actress Oscar at age 25. The mean age of best supporting actor is 50.1 with a
SD of 13.9. The mean age for best supporting actress is 39.7 with a SD of 14. Find the z-score
that corresponds to the age for each person and compare your results.