Looking at Data-Distributions
1.1-Displaying Distributions with
Graphs
Basic definitions
• Data-numbers with a context
 Eg. Your friends new baby weighed 10.5 pounds, we know that baby is
quite large. But if it is 10.5ounces or 10.5kg, we know that it is
impossible-the context makes the number informative
• Individuals-objects described in the data(people,animals,things)
• Variable-any property/characteristics of an individual(IQ scores of persons)
• Distribution-of a variable tells us what values & how often(frequency of a
variable)
Types of variables
• categorical variable-places an individual into
one of several categories(male/female,
smoker/nonsmoker)
• quantitative variable-takes numerical values
for which arithmetic operations such as
adding & averaging can be performed(shoe
size,age)
How to represent data?
• Categorical variables-can use Pie-chart & bar
graphs
 Eg. make a pie chart/bar graph for distribution of gender
• Quantitative variables-can use histogram
Example 1-The color of your car(distribution of the most popular colors for
2005 model luxury cars made in North America
Color
Percent
Silver
20
White, pearl
18
Black
16
Blue
13
Light brown
10
Red
7
Yellow,gold
6
a)
b)
c)
d)
What percent of vehicles are some other color?
Make a bar graph?
Can we make a pie chart for the given colors?
Would it be correct to make a pie chart if you added an “Other”
category?
Example 2-The density of the earth (the variable recorded was the density of
the earth as multiple of the density of water)
• Using TI-84 create a histogram
• Discuss the shape, center, spread and outliers
Example3-Do women study more than men?
Variable-minutes studied on a typical weeknight of a first-year college class
•
•
Here are the responses of random samples of 30 women and 30 men from the class:
Women 180,120,150,200,120,90,120,180,120,150,60,240,180,120,180,180,120,
180,360,240,180,150,180,115,240,170,150,180,180,120
• Men
90,90,150,240,30,0,120,45,120,60,230,200,30,30,60, 120, 120, 120,
90,120,240,60,95,120,200,75,300,30,150,180
a) Examine the data. Why are you not surprised that more responses are multiples of
10minutes? We eliminated one student who claimed to study 30,000 minutes per
night. Are there any other responses you consider suspicious?
b) Make a back-to-back stem plot to compare the two samples. That is, use one set of
stems with two sets of leaves, one to the right and one to the left of the stems.(Draw
a line on either side of the stems to separate stems and leaves.) Order both sets of
leaves from smallest at the stem to largest away from the stem. Report the
approximate midpoints of both graphs. Does it appear that women study more than
men(or at least claim that they do)?
Answers
• a) Most people round their answers. The students who claimed 0 minutes,
360 minutes and 300 minutes.
• B)The stemplots suggest that women(claim to) study more than men. The
approximate centers are 175 minutes for women and 120 minutes for
men.
Looking at Data-Distributions
1.2-Describing Distributions with
numbers
Mean & Median
• Mean =sum of numbers/ number of numbers
• Median=Middle value(when the numbers are in ascending order)
– Example 1: 103,105,109,140,170 (Median is 109-the number in the (n+1)/2th position
from the bottom of the list-n is number of values)
– Example 2: 18,19,20,20,26,28(Median is 20- the avg of n/2 position number & n/2+1
position). Mean =21.83
– Example 3: replace 28 in example 2 by 100 & re -compute mean and median?
•
•
•
18,19,20,20,26,100
Mean =33.83
Median-does not change
Mean is affected by outliers
Median is not affected by outliers
A measure of center alone can be misleading
Solution-need a measure of spread(variability)
Measuring spread
Quartiles
• Example 4–Age of 10 students
• 26,19,20,18,20,19,19,19,19,21
• Sort them in ascending order
• 18,19,19,19,19,19,20,20,21,26
• Median =19 (Q2 )
• First quartile=median of the lower half of data(Q1 )=19
• Third quartile=median of the upper half of data(Q3 )=20
Five-number summary
• Min Q1 Q2 Q3 Max
• Box plot- Picture of the five number summary. Can be used to compare two
distributions
Max
Q3
IQR
Median(Q2 )
Q1
Min
• IQR(Inter quartile range)= Q3 - Q1
The 1.5 X IQR rule for suspected outliers
• Example 5(travel times to work in New York-in minutes)
• 10,30,5,25,40,20,10,15,30,20,15,20,85,15,65,15,60,60,40,45
• (single peaked/right skewed/no center observation,but there is a center
pair)
• The five number summary
• 5 15 22.5 42.5 85
• IQR=42.5-15=27.5
• Apply 1.5XIQR rule
– Step 1:calculate 1.5 X IQR=1.5 x 27.5
–
–
–
–
Step 2: Calculate Q1 -(1.5 X IQR)= 15-41.25=-26.25
Step 3: Calculate Q3 +(1.5 X IQR)=42.5+41.25=83.75
Any values outside of (-26.25,83.75) are flagged as outliers
The suspected outlier in the data is 85
Standard deviation(s)
•
•
•
•
•
•
•
Used as a measure of spread when mean=center
Units of s=same as data units
s always positive
Higher s->more spread
s=0->no spread -> all observations equal
s affected by outliers
Example :1,1,2,5,3
Formulas for variance & standard deviation
Variance
Standard deviation
Looking at Data-Distributions
1.3 –Density Curves and Normal
Distributions
Definitions
• Density Curve-Special type of histogram such that total area under the
curve is 1
Relative
frequency
• Typical histogram
• Example for a Density Curve
adjusted relative frequency
Density Cuve-Histogram with total area under line
curve is 1
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-40.9-32.9-24.9-16.9 -8.9 -0.9 7.1 15.1 23.1 31.1 39.1
Bin mid points
Bin limits
Characteristics of density curve
•All y values positive
• total area under curve=1
•Curve approaches to zero for
extreme left & right x values
Definitions
Normal Distribution
• Formula
It can be shown that the probability density function for a normal random
variable, X, with mean X and standard deviation X has the following
 x 
form.

0.5
1
2
X
fX ( x ) 
 X  2 
e
 X 
• TI-84 calculator-> 1)STAT plot off 2) enter in Y1-use
normalpdf(x,mean,standard deviation) 3)normalpdf( ) found in 2nd->DISTR
Normal density curve-Mean 0 &
Standard deviation 1
Normal density curve-Mean 6 &
Standard deviation 2
0.6
0.6
0.4
0.4
f(x)
f(x)
-4
-2
0.2
0.2
0
0
0
x
2
4
-4
-2
0
x
2
4
Definitions
The 68-95-99.7 rule
1. Example-When Mean 0 & standard deviation is 1
•
•
•
Approximately 68% of the observations fall within one standard deviation of
the mean
Approximately 95% of the observations fall within two standard deviation of
the mean
Approximately 99.7% of the observations fall within three standard deviation
of the mean
Definitions
• tables-allows us to calculate the probabilities for a
normal distribution
• How to get numbers?
– There are too many normals(one per possible mean/one
per possible standard deviation)
– >infinitely many
• Need to standardize
– Standardization of Normal Random Variables. If X is
normally distributed, its standardization is
– Equation:
Definitions
• Standard normal(Z) : N(0,1) , mean 0 &
Standard deviation 1
• Now can calculate the fraction of my data set
between any two limits