1. Measures of Center and Variation Sections 3.1 and 3.3
Prof. Felix Apfaltrer
Office:N518
Phone: X 7421
Office hours:
Mon-Thu 1:30-2:15 pm

2. Measures of center - mean
A measure of center is a value that represents the center of the data set
The mean is the most important measure of center (also called arithmetic mean)
sample mean
population mean
addition of values
variable (indiv. data vals)
sample size
population size
Example. Lead (Pb) in air at BMCC (mmg/m3), 1.5 high:
5.4, 1.1, 0.42, 0.73, 0.48, 1.1
Outlier has strong effect on mean!

3. Measures of center - median
Mean is good but sensitive to outliers!
Large values can have dramatic effect!
The median is the middle value of the original data arranged in increasing order
If n odd: exact middle value
If n even: average 2 middle values
Previous example:
reorder data:
0.42, 0.48, 0.73, 1.1, 1.1, 5.4
If we had an extra data point:
5.4, 1.1, 0.42, 0.73, 0.48, 1.1, 0.66
After reordering we have
0.42, 0.48, 0.66, 0.73, 1.1, 1.1, 5.4
Outlier has strong effect on mean, not so on median!
Used for example in median household income: $ 36,078

4. Measures of Center - mode and midrange
Mode M value that occurs most frequently
if 2 values most frequent: bimodal
if more than 2: multimodal
Iif no value repeated: no mode
Needs no numerical values
Midrange
= (highest-lowest value)/2
Outliers have very strong weight
Examples:
5.4, 1.1,0.42, 0.73, 0.48, 1.1
27, 27, 27, 55, 55, 55, 88, 88, 99
1, 2, 3, 6 , 7, 8, 9, 10
Solutions:
unimodal: 1.1
Bimodal 27 and 55
No mode
a. ( )/2=2.91
b. (27+99)/2=63
c. (1+10)/2= 5.5

5. Mode and more … Mode: not much used with numerical data Weighted mean:
Example:
Survey shows students own:
84% TV
76% VCR
69% CD player
39% video game player
35% DVD
Mean from frequency distribution
Weighted mean:
Dis-Advantages of different measures of center
TV is the mode!
No mean, median or midrange!
Round-off: carry one more decimal than in data!

6. Measures of variation Variation measures consistency
Range = (highest value - lowest value)/2
Standard deviation:
Precision
arrows
jungle
arrows
Same mean length, but different variation!

7. Standard deviation Measure of variation of all values from mean
Recipe:
Compute the mean
Substract mean from
Individual values
Square the differences
Add the squared differences
Divide by n-1.
Take the square root.
Example: waiting times
Bank Consistency
Bank Unpredictable
Mean: ( )/6=5
(6-5)=1,(5-5)=0, (4-5)=-1, (4-5)=-1, (6-5)=1, 0
12=1 , 02=0, (-1)2=1, (-1)2=1, 12=1,02=0
∑ = 4
n-1=6-1= /5=0.8
√0.8 = 0.9 min vs min
Measure of variation of all values from mean
Positive or zero (data = )
Larger deviations, larger s
Can increase dramatically with outliers
Same units as original data values

8. Standard deviation of sample and population
Standard deviation of a population
divide by N
- mu (population mean)
Sigma (st. dev. of population)
Different notations in calculators
Excell: STDEVP instead of
STDEV
Example using fast formula:
Find values of n, ,
n=6 6 values in sample
= 30 adding the values
= = 154
Estimating s and  :
(highest value - lowest value)/4

9. Example: class grades
A statistics class of 20 students obtains the following grades:
To rapidly approximate the mean, we take a random sample of 5 students. At random, we pick
x = ( )/5=395/5 =79
s =√((78-79) 2 +(92-79) 2 +(64-79)2+(83-79) 2 +(78-79)2)/4
=√(( -1) ( 13 ) 2 + ( -15 )2+ ( 4 ) 2 +( -1 )2)/4
=√( )/4
=√( 412 )/4 =√( 103 ) = 10.15
The population mean is obtained
by adding all grades
and dividing by 20, which is
The population variance is
Which we can obtain using Excell:

10. Variance and coefficient of variation
Variance = square of standard deviation
sample
population
General terms refering to variation: dispersion, spread, variation
Variance: specific definition
Ex: finding a variance 0.8, 40
Examples:
In class grade case, sample standard deviation was
Therefore, s2=103.
The population standard deviation was 10.71, therefore,
 2= =

11. Coefficient of variation
Coefficient of variation CV [p.155 ex. 49]
Describes the standard deviation relative to the mean:
Coefficient of variation allows to compare dispersion of completely different data sets
ex:
consistent bank data set
6,5,4,4,6,5; x=5, s=0.9
CV=.9/5=0.18
Class sample: x=79, s=10.1
CV=10.1/79=0.13
Variation of consistent bank is larger than that of the class in relative terms!
In previous example,
CVsample=10.1/79 =12.8%
CVpopulation=10.71/ =13.4%

12. More on variance and standard deviation
Empirical rule for data with normal distribution
Why use variance, standard deviation is more intuitive?
(Independent) variances have additive properties
Probabilistic properties
Standard deviation is more intuitive
Why divide sample st. dev by n-1?
Only n-1 free parameters
68% of data
95% of data
99.7% of data
Example: Adult IQ scores have a bell-shaped distribution with mean of 100 and a standard deviation of 15. What percentage of adults have IQ in 55:145 range?
s=15, 3s=45, x-3s=55, x+3s=145 Hence, 99.7% of adults have IQs in that range.
Chebyshev’s theorem: At least 1-1/k2 percent of the data lie between k standard deviations from the mean. Ex: At least 1-1/3^2=8/9=89% of the data lie within 3 st. dev. of the mean.

13. The mean and the median are often different
This difference gives us clues about the shape of the distribution
Is it symmetric?
Is it skewed left?
Is it skewed right?
Are there any extreme values?

14. Symmetric – the mean will usually be close to the median
Skewed left – the mean will usually be smaller than the median
Skewed right – the mean will usually be larger than the median
Skewness: Pearson’s index
I=3( mean-median )/s
If I < -1 or I > 1: significantly skewed

15. For a mostly symmetric distribution, the mean and the median will be roughly equal
Many variables, such as birth weights below, are approximately symmetric

16. Summary: Chapter 3 – Sections 1and 2
Mean
The center of gravity
Useful for roughly symmetric quantitative data
Median
Splits the data into halves
Useful for highly skewed quantitative data
Mode
The most frequent value
Useful for qualitative data
Range
The maximum minus the minimum
Not a resistant measurement
Variance and standard deviation
Measures deviations from the mean
Empirical rule
About 68% of the data is within 1 standard deviation
About 95% of the data is within 2 standard deviations

17. Summary: Chapter 3 – Section 3 (Grouped Data)
As an example, for the following frequency table,
we calculate the mean as if
The value 1 occurred 3 times
The value 3 occurred 7 times
The value 5 occurred 6 times
The value 7 occurred 1 time
Class
0 – 1.9
2 – 3.9
4 – 5.9
6 – 7.9
Midpoint 1 3 5 7
Frequency 6

18. Evaluating this formula
The mean is about 3.6
In mathematical notation
This would be μ for the population mean and for the sample mean

19. Variance and Standard deviation (grouped data)
Interpreting a known value of the standard deviation s: If the standard deviation s is known, use it to find rough estimates of the minimum and maximum “usual” sample values by using
max “usual” value ≈ mean + 2(st. dev)
min “usual” value ≈ mean - 2(st. dev)
Finding s from a frequency distribution
Example: cotinine levels of smokers
N-1: DATA 3,6,9
=6,  2=6
Samples (replacement):
x =
∑(x-x )2 =
S2=(divide by n-1=2-1)
Mean value of s2= /9 = 6
S 2=(divide by n=2)
Mean value of s 2= /9 = 3
using Excel we obtain
with which we calculate:

20. Measures of relative standing
Useful for comparing different data sets
z scores
Number of standard deviations that a value x is above of below the mean
Percentiles:
Percentile of value x Px
total number of values
Px=
number of values less than x
sample population
Example
data point 48 in Smoker data
8/40*100=20th percentile = P20
Exercise:
Locate the percentiles of data points 1, 130 and 250.
Example:
NBA Jordan 78, =69,  =2.8
WNBA Lobo 76, =63.6,  =2.5 Number of standard deviations that a value x is above of below the mean
J: z=(x-)/=(78-69)/2.8=3.21
L: z=(x-)/=( )/2.5=4.96

21. Quartiles and percentiles

22. Percentiles and Quartiles
Yes:
take average of
Lth and (L+1)st value
as Pk
No:
ROUND UP
Pk is the Lth value
Compute
L=(k/100)*n
n=number of values
k=percentile
SORT DATA
START
L whole
number?
total number of values
Pk: k=
number of values less than x
Quartiles:
Q1,= P25, Q2 = P50 =median, Q3= P75
Pk: k = (L – 1)/n •100
Example: data point 48 in Smoker data is 9th on table, n= 40.
(9 – 1)/40 •100=20  48 is in P20 or 20th percentile or the first quartile Q1.
Data point 234 is 28th. k=(28 – 1)/40 •100= 68th percentile, or the 3rd quartile Q3.
Example: In class table ( n = 20 )
find value of 21 percentile
L=21/100 * 20 = 4.2
round up to 5th data point
--> P21 = 71
find the 80th percentile:
L=80/100 * 20 = 16,
WHOLE NUMBER:
P80 =(89+92)/2=90.5
Conversely, if you are looking for data in the kth percentile:
L=(k/100)*n
n total number of values
k percentiles being used
L locator that gives position of a value
(the 12th value in the sorted list L=12)
Pk kth percentile (ex: P25 is 25th percentile)

23. Exploratory Data Analysis
Exploratory data analysis is the process of using statistical tools (graphs, measures of center and variation) to investigate data sets in order to understand their characteristics.
Box plots have less information than histograms and stem-and-leaf plots
Not that often used with only one set of data
Good when comparing many different sets of data
Outlier: Extreme value. (often they are typos when collecting data, but not always).
can have a dramatic effect on mean
can have dr. effect on standard deviation
… on histogram