Goals
Notes
Order Statistics: Measures of Positions
§2.5 Goals:
I
Compute quartiles.
I
Graph position using quartiles.
I
Standardizing data.
I
Computing outliers.
For June 9:
I
Homework §2.5: #44, #51, #56, #57 (by hand)
Suggested Exercises: §2.5: #4, #6, #7, # 13, #14
Quartiles
Notes
Three quartiles roughly divide an ordered data set into four equal
parts. About 14 of the data fall below the first quartile Q1 . About half
of the data falls below the second quartile, Q2 . About 34 of the data
falls below the third quartile.
Ex: Data Set: 15 CPR Training Test Scores:
5
13
18
7 9 10 11
14 15 16 17
18 20 21 27
5
13
18
7 9 10 11
14 15 16 17
18 20 21 27
Interquartile Range
Notes
The measure of variation giving the range of the middle 50% of the
data is the distance measured by the difference of Q3 and Q1 .
IQR = Q3 − Q1
Ex: For CPR data, IQR = 18 − 10 = 8. Think of this as distance
from the number 10 to 18.
Ex: Tail lengths of a sample of American alligators in feet:
3.4
6.5
3.9 4.2 4.6
6.8 7.1 7.5
5.4
Q1 = (3.9 + 4.2)/2 = 4.05 feet, Q2 = 5.4 feet,
Q3 = (6.8 + 7.1)/2 = 6.95 feet. IQR = 6.95 − 4.05 = 2.9 feet.
Five-Number Summary and Boxplots
Notes
The minimum data entry, Q1 , Q2 , Q3 , and maximum data entry
together are called the five-number summary. Drawing a
box-and-whisker plot (or box plot) is a descriptive statistics tool that
highlights these features of the data set.
Ex: For the CPR test scores:
1. The minimum is 5.
2. Q1 = 10
3. Q2 = 15
4. Q3 = 18
5. The maximum is 27.
Five-Number Summary and Boxplots
Five-Number Summary and Boxplots
Give the five-number summary for the alligator tail lengths. Draw the
box plot.
Notes
Notes
z-score
Notes
A data value has a standard score or z-score that represents position
from the mean measured by multiples of the standard deviation.
Ex: A statistics test has a mean of µ1 = 63 and standard deviation
σ1 = 7.0. A biology test has a mean of µ2 = 23 and standard
deviation σ2 = 3.9.
Student A scores a 75 on the statistics test. Student B scores a 25 on
the biology test. Which student earned a higher standard score?
Unusual z-scores
Notes
The Empirical Rule is applied to approximately bell-shaped
distributions. We know that about 95% of the data lie within two
standard deviations of the mean. Unusual values lie outside of two
standard deviations. Data can be transformed into z-scores, and
about 95% of the data lie within -2 and 2.
Unusual z-scores
Monthly utility bills 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. Are any of these bills unusual? Explain.
Notes