Download Measures of Spread

Exploring Data
1.1 Displaying Distributions with Graphs
Mrs. Slobig
•
Does the distribution have one or more peaks
(modes) or is it unimodal?
•
Is the distribution approximately symmetric or is
it skewed in one direction? Is it skewed to the
right (right tail longer) or left?
Example Description
•
Shape: The distribution is roughly symmetric
with a single peak in the center.
•
Center: You can see from the histogram that
the midpoint is not far from 110. The actual
data shows that the midpoint is 114.
•
Spread: The spread is from 80 to about 150.
There are no outliers or other strong deviations
from the symmetric, unimodal pattern.
Calculator Example
Text
(To save data for later use on home screen type L1 -> Prez)
Calc continued
•
Frequency shortcut: If you have a dataset
comprised of 75 3’s and 35 4’s for example,
you can enter the values in list 1 and the
frequencies in list 2 then pull 1 variable stats:
•
Stats-edit- L1: 3, 4
L1: 75, 35
stat-calc-1var stats L1,L2 enter
Relative
frequency/Cumulative
Frequency
•
A histogram does a good job of displaying the
distribution of values of a quantitative variable, but tells
us little about the relative standing of an individual
observation.
•
So, we construct an ogive (“Oh-Jive”) aka a relative
cumulative frequency graph.
Step 1- Construct table
•
Decide on intervals and make a frequency table with 4 columns: Freq,
Relative frequency, cumulative frequency, and rel. cum. Freq.
•
To get the values in the rel. freq. column, divide the count in each
class interval by the total number of observations. Multiply by 100
to convert to %.
•
In Cum freq column, add the counts that fall in or below the current
class interval
•
for rel. cum. freq. column, divide the entries in the cum freq column
by total number of individuals.
Step 2 & 3
•
Label and scale your axes and title your graph.
Vertical axis always Relative Cum. Freq. Scale
the horizontal axis according to your choice of
class intervals and the vertical axis from 0% to
100%.
•
Plot a point corresponding to the rel. Cum. freq.
in each class interval at the LEFT ENDPOINT
of the NEXT class interval. (example, the 40 to
44 interval, plot a point at a height of 4.7%
above the age value of 45.
•
Begin with 0% you should end with 100%.
Connect dots
To Locate an individual within distribution: What about Clinton? He was 46. To find his relative standing, draw a vertical line up from
his age (46) on the horizontal axis until it meets the ogive. Then draw a horizontal line from this point of intersection to the vertical
axis. Based on our graph his age places him at the 10% mark which tells us that about 10% of all US presidents were the same age
as or younger than Bill Clinton when they were inaugurated.
To locate a value corresponding to a percentile, do the opposite. Ex: 50th percentile, 55 years old.
•
Whenever data are collected over time, plot
observations in time order. Displays of
distributions such as stemplots and histograms
which ignore time order can be misleading
when there is systematic change over time.
Shows change in gas price over time. Shows TRENDS
Exploring Data
1.2 Describing Distributions with Numbers
Mrs. Slobig
Sample Data
Consider the following test scores for a small class:
75
76
82
93
45
68
74
82
91
Plot the data and describe the SOCS:
Shape?
Outliers?
Center?
Spread?
What number best describes the “center”?
What number best describes the “spread’?
98

Measures of Center
Numerical descriptions of distributions begin with a
measure of its “center”.
If you could summarize the data with one number,
what would it be?
Mean:
x
The “average” value of a dataset.
x1  x2  ... xn
x
n
x

x
i
n
Median: Q2 or M The “middle” value of a dataset.
Arrange observations in order min to max

Locate the middle observation, average if needed.
Mean vs. Median
The mean and the median are the most common
measures of center.
If a distribution is perfectly symmetric, the mean
and the median are the same.
The mean is not resistant to outliers.
You must decide which number is the most
appropriate description of the center...
Measures of Spread
Variability is the key to Statistics. Without variability,
there would be no need for the subject.
When describing data, never rely on center alone.
Measures of Spread:
Range - {rarely used...why?}
Quartiles - InterQuartile Range {IQR=Q3-Q1}
Variance and Standard Deviation {var and sx}
Like Measures of Center, you must choose the most
appropriate measure of spread.
Quartiles
Quartiles Q1 and Q3 represent the 25th and 75th
percentiles.
To find them, order data from min to max.
Determine the median - average if necessary.
The first quartile is the middle of the ‘bottom half’.
The third quartile is the middle of the ‘top half’.
19
22
23
23
23
Q1=23
45
68
74
Q1
26
26
27
28
med
75
76
82
med=79
29
30
31
32
Q3=29.5
82
91
Q3
93
98
5-Number Summary, Boxplots
The 5 Number Summary provides a reasonably
complete description of the center and spread of
distribution
MIN
Q1
MED
Q3
MAX
We can visualize the 5 Number Summary with a
boxplot.
min=45
45
50
55
Outlier?
Q1=74
med=79
60
70
65
75
Quiz Scores
Q3=91
80
85
max=98
90
95
100
Determining Outliers
“1.5 • IQR Rule”
InterQuartile Range “IQR”: Distance between Q1 and
Q3. Resistant measure of spread...only measures
middle 50% of data.
IQR = Q3 - Q1 {width of the “box” in a boxplot}
1.5 IQR Rule: If an observation falls more than 1.5
IQRs above Q3 or below Q1, it is an outlier.
Why 1.5? According to John Tukey, 1 IQR seemed like
too little and 2 IQRs seemed like too much...
1.5 • IQR Rule
To determine outliers:
Find 5 Number Summary
Determine IQR
Multiply 1.5xIQR
Set up “fences” Q1-(1.5IQR) and Q3+(1.5IQR)
Observations “outside” the fences are outliers.
Outlier Example
All data on
p. 48.
IQR=45.72-19.06
IQR=26.66
1.5IQR=1.5(26.66)
1.5IQR=39.99
fence: 19.06-39.99
= -20.93
fence: 45.72+39.99
= 85.71
outliers
}
{
0
10
20
30
40
50 60 70
Spending ($)
80
90
100
Standard Deviation
Another common measure of spread is the Standard
Deviation: a measure of the “average” deviation of all
observations from the mean.
To calculate Standard Deviation:
Calculate the mean.
Determine each observation’s deviation (x - xbar).
“Average” the squared-deviations by dividing the
total squared deviation by (n-1).
This quantity is the Variance.
Square root the result to determine the Standard
Deviation.
Standard Deviation
Variance:
(x1  x ) 2  (x2  x ) 2  ... (xn  x ) 2
var 
n 1
Standard Deviation:

sx 
2
(x

x
)
 i
n 1
Example 1.16 (p.85): Metabolic Rates
1792
1666
1362

1614
1460
1867
1439
Standard Deviation
1792
1666
1362
1614
1460
1867
1439
Metabolic Rates: mean=1600
x
(x - x)
(x - x)2
1792
192
36864
1666
66
4356
1362
-238
56644
1614
14
196
1460
-140
19600
1867
267
71289
1439
-161
25921
Totals:
0
214870
Total Squared
Deviation
214870
Variance
var=214870/
6
var=35811.66
Standard
Deviation
s=√35811.66
s=189.24 cal
What does this
value, s, mean?
Linear Transformations
Variables can be measured in different units (feet vs
meters, pounds vs kilograms, etc)
When converting units, the measures of center and
spread will change.
Linear Transformations (xnew=a+bx) do not change
the shape of a distribution.
Multiplying each observation by b multiplies both
the measure of center and spread by b.
Adding a to each observation adds a to the
measure of center, but does not affect spread.
Data Analysis Toolbox
To answer a statistical question of interest:
Data: Organize and Examine
Who are the individuals described?
What are the variables?
Why were the data gathered?
When,Where,How,By Whom were data gathered?
Graph: Construct an appropriate graphical display
Describe SOCS
Numerical Summary: Calculate appropriate center and
spread (mean and s or 5 number summary)
Interpretation: Answer question in context!
Chapter 1 Summary
Data Analysis is the art of describing data in context
using graphs and numerical summaries. The
purpose is to describe the most important features of
a dataset.