AP STAT: CHAPTERS 4 & 5
QUANTITATIVE DATA
Quantitative Distributions:
1. Dotplots
positives: see all observations
Example:
# of siblings of classmates
negatives: not good for large sets of data
0
# of siblings of classmates
2. Stemplot (aka Stem and Leaf Plot)
* no commas * leaves are single digits
Example:
Babe Ruth’s homerun totals each season for the Yankees:
54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 41, 34, 22
Babe Ruth’s HR totals
2
2 5
3
4 5
4
1 1 6 6 7 9
Key:
5
4 4 9
2 2 = 22 homeruns
6
0
3. Back to Back Stemplot:
Example:
Roger Maris’ homerun totals for the Yankees: 08, 13, 23, 33, 28, 16, 14, 39, 26, 61
Create a Back-to-Back stemplot with Babe Ruth’s
Roger Maris’ HR’s
Babe Ruth’s HR’s
8
0
6 4 3
1
8 6 3
2
2 5
9 3
3
4 5
4
1 1 6 6 7 9
5
4 4 9
Key:
1
6
0
0 8 = 8 homeruns
outlier
Splitting Stems: When?
When your data is clumped together within only a few stems
In what ways can the stems be split?
By 5, 2, 1
Age guess example…many people guessed ages in the 20’s and 30’s and so the data was clumped
together
4. Histograms
Example:
*don’t see every piece of data
#
Or
%
bins= bar width
* must use a minimum of 5 bars
Histograms on the calculator: see page 46 in the book for help
Example: The following are a list of test scores on an exam. Create a histogram of these scores.
variable
40
61
66
68
74
77
84
91
76
82
87
53
64
57
51
67
81
65
45
62
66
69
75
78
84
95
76
82
90
67
71
59
64
70
85
68
49
64
66
69
76
80
85
96
64
68
72
76
81
86
99
76
96
72
BINS = 6
x-min = 40
x-max = 100
x scale (bar width) = 10
#
Test Score (%)
4 types of histograms:
Frequency (calc)
#
Relative Frequency
%
Cumulative Frequency
#
#
Cumulative Relative Frequency
%
Histogram Examples:
Example 7:
Examples 8:
Using the list INCOM, create a frequency histogram.
Using the list GPA, create a relative frequency histogram.
Describing Distributions (numerically):
Center: MEDIAN - middle observation – 50 % above/below
QUARTILES – medians of the lower and upper half of the data
Q3 = median of upper half --- 75% of data falls below
Q1 = median of lower half --- 25% of data falls below
Spread: RANGE - (min, max)
IQR – Q3 - Q1 ---- middle 50% of the data
5# Summary:
Minimum
1st Quartile (Q1)
Median (Q2)
3rd Quartile (Q3)
Maximum

Center: MEAN – arithmetic average
Sigma (sum)
FORMULA:
x
x
On the calc  1-var stats (page 65 in book)
individual piece of data
i
n
total
* read as “x-bar”
* on calculator: 1-var stats…
Spread: STANDARD DEVIATION & RANGE
Sx
(a, b)
Standard deviation (s) is: a number that describes how much the data varies (or how spread out it is)
* average difference of each point from the mean
* the higher the number, the more spread out the data is
Examples:
s = 10
s=2
Formula:
n
s
 ( y  y)
i 1
2
i
n 1
On calculator: STAT  CALC  1 Var Stats
mean
standard
deviation
Properties:
* s = 0 when all data is the same
* Unless all point are the same, “s” is always POSITIVE
*Iit is not resistant! It is affected by outliers
* Variance = Standard Deviation square
OR
Ex: Variance = 16 …. Standard Deviation = 4
Standard Deviation = square root of variance
Back to Quantitative Distributions:
4) Boxplots
Graphical display of 5 number summary
minimum
Q1
Median (Q2)
Outliers: 1.5 x IQR rule
Lower Fence (LF) = Q1  1.5( IQR)
Written: (LF, UF)
Upper Fence (UF) = Q3  1.5( IQR)
Example: Using the AGES in months data, test to see if there are any outliers
Q3
maximum
Modified Boxplot
If outliers are present the whisker extends to the lat value that is not an outlier
- The outlier is marked with an x, box, or point
outiers
Parallel Boxplots:
2 or more boxplots on the same scale
* used to compare two or more similar variables
On the calculator, compare the lists SATMF and SATMM which are the SAT math scores for males and females
DESCRIBING DISTRIBUTIONS: Shape, Center, Spread
SHAPE
MODE:
UNIMODAL
BIMODAL
SHAPES:
UNIFORM
SYMMETRIC
LEFT SKEWED
RIGHT SKEWED
OUTLIERS
GAPS
CLUSTERED
GRANULARITY
OTHER:
* consistent, understandable gaps
CENTER:
MEDIAN: used when comparing skewed distributions or distributions with outliers
MEAN (average): used when comparing symmetric or roughly symmetric distributions
SPREAD:
* goes along with what measure of center you choose
Mean  standard deviation & range
Median  IQR & range
Mean vs. Median:
SYMMETRIC:
mean  median
mean
median
mode
LEFT SKEWED:
mean < median
median
mode
mean
RIGHT SKEWED:
mean > median
median
mean
mode
*the mean gets pulled towards the tail in the direction of the skew
EXAMPLES (worksheet)
Comparing 2 (or more) distributions
* Need to mention shape, center, and spread of each
* Must make a comparison of the shape, center, and spread
- give specific numerical data
- use comparison words like greater than, less than, higher than, lower than, the same as, equal to,
different from
- each thing can be compared in one sentence
Example: The median for Block 1 was 76 points which was higher than the median of Block 2 which was 73 points.
When comparing:
Symmetric to symmetric : Use the mean, standard deviation, and range
Skewed to skewed: Use the median, IQR, and range
Skewed to symmetric: Use the median, IQR, and range
CANNOT… Compare Mean to Median
Example: Compare and describe the SATMF and SATMM data sets
Shape: The males are unimodal with no outliers and right skewed which is different to the females which are
unimodal with no outliers yet left skewed.
Center: The median for males is 610 points which is lower than the median for females which is 700 points.
Spread: The IQR for males is 55 points which is lower than the IQR for females which is 120 points. The range for
males is (480, 700) which is smaller than the range for females which is (500, 800).