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### Measures of Center and Variation Sections 3.1 and 3.3

Prof. Felix Apfaltrer

fapfaltrer@bmcc.cuny.edu

Office:N518

Phone: 212-220 8000X 7421

Office hours:

Mon-Thu 1:30-2:15 pm

Measures of center - mean

addition of values

variable (indiv. data vals)

sample size

population size

- 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

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!

Measures of center - median

Previous example:

- reorder data:

0.42, 0.48, 0.73, 1.1, 1.1, 5.4

- 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 nodd: exact middle value
- If n even:average 2 middle values

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

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

Measures of Center - mode and midrangeExamples:

- 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. (0.42+5.4)/2=2.91

b. (27+99)/2=63

c. (1+10)/2= 5.5

Mode: not much used with numerical data

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

Mode and more …TV is the mode!

No mean, median or midrange!

Round-off: carry one more decimal than in data!

Measures of variation

- Variation measures consistency
- Range = (highest value - lowest value)/2
- Standard deviation:

Precision

arrows

jungle

arrows

Same mean length, but different variation!

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

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 6 5 4 4 6 5

Bank Unpredictable 0 15 5 0 0 10

Mean: (6+5+4+4+6+5)/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

∑ 1+0+1+1+1+0 = 4

n-1=6-1=5 4/5=0.8

√0.8 = 0.9 min vs 6.3 min

Standard deviationExample using fast formula:

Find values of n, ,

n=6 6 values in sample

= 30 adding the values

= 62+52+42 +42 +52+ 62= 154

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

Standard deviation of sample and populationEstimating s and :

(highest value - lowest value)/4

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= (78+92+64+83+78)/5=395/5 =79

s =√((78-79) 2 +(92-79) 2 +(64-79)2+(83-79) 2 +(78-79)2)/4

=√(( -1) 2 + ( 13 ) 2 + ( -15 )2+ ( 4 ) 2 +( -1 )2)/4

=√( 1 + 169 + 225 + 16 + 1)/4

=√( 412)/4 =√( 103) = 10.15

The population mean is obtained

by adding all grades

and dividing by 20, which is 79.95.

The population variance is 10.71.

Which we can obtain using Excell:

Example: class gradesVariance

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 10.15.

Therefore, s2=103.

The population standard deviation was 10.71, therefore,

2=10.71 2= 114.7.

Variance and coefficient of variationCoefficient 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!

Coefficient of variation CV [p.155 ex. 49]

Describes the standard deviation relative to the mean:

Coefficient of variationIn previous example,

CVsample=10.1/79 =12.8%

CVpopulation=10.71/ 79.95 =13.4%

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

Empirical rule for data with normal distribution

68% of data

95% of data

99.7% of data

More on variance and standard deviationExample: 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.

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?

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

For a mostly symmetric distribution, the mean and the median will be roughly equal

- Many variables, such as birth weights below, are approximately symmetric

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
- Not a resistant measurement
- Empirical rule
- About 68% of the data is within 1 standard deviation
- About 95% of the data is within 2 standard deviations

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

- The mean is about 3.6
- In mathematical notation
- This would be μ for the population mean and for the sample mean

Finding s from a frequency distribution

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)

Variance and Standard deviation (grouped data)Example: cotinine levels of smokers

N-1: DATA 3,6,9

=6, 2=6

Samples (replacement): 33 36 39 63 66 69 93 96 99

x = 3 4.5 6 4.5 6 7.5 6 7.5 9

∑(x-x )2 = 0 4.5 18 4.5 0 4.5 18 4.5 0

S2=(divide by n-1=2-1) 0 4.5 18 4.5 0 4.5 18 4.5 0

Mean value of s2=54/9 = 6

S2=(divide by n=2)0 2.25 9 2.25 0 2.25 9 2.25 0

Mean value of s2=27/9 = 3

using Excel we obtain

with which we calculate:

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

sample population

number of values less than x

Px=

total number of values

Measures of relative standingExample

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-)/=(76-63.6)/2.5=4.96

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)

Quartiles:

Q1,= P25, Q2 = P50 =median, Q3= P75

START

number of values less than x

Pk: k=

SORT DATA

total number of values

Compute

L=(k/100)*n

n=number of values

k=percentile

Yes:

L whole

number?

take average of

Lth and (L+1)st value

as Pk

No:

ROUND UP

Pk is the Lth value

Percentiles and QuartilesPk: 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

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

Exploratory Data AnalysisOutlier: 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

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