Probabilistic & Statistical Techniques

Probabilistic & Statistical Techniques

Eng. Tamer Eshtawi

First Semester 2007-2008

Eng. Tamer Eshtawi

First Semester 2007-2008

Lecture 5

Chapter 2 (part 3)

Statistics for Describing

DataMain Reference: Pearson

Education, Inc Publishing as Pearson Addison-Wesley.

Section 3-4Measures of position

Key Concept

This section introduces measures that can be used to compare values from different data sets, or to compare values within the same data set. The most important of these is the concept of the z score.

z Score (or standardized value)the number of standard

deviations that a given value x is above or below the mean

Definition

Sample

x

z

Population

Round z to 2 decimal places

Measures of Position z score

s

xxz

Interpreting Z Scores

Whenever a value is less than the mean, its corresponding z score is negative

Ordinary values: z score between –2 and 2 Unusual Values: z score < -2 or z score > 2

Definition

Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%.

Q1 (Third Quartile) separates the bottom 75% of sorted values from the top 25%.

Q1, Q2, Q3 divide ranked scores into four equal

parts

Quartiles

25% 25% 25% 25%

Q3Q2Q1(minimum) (maximum)

(median)

Find lower & upper Quartile

To fined Q1, first calculate one-quarter of n and add ½ to obtain ¼ n + ½ . Round this to nearest integer.

Example 1 1 1 2 3 3 8 11 14 19 19 20

n = 11,then ¼ n + ½ = ¼ (11)+½ = 3.25 rounded off to 3Q1 = 2Q3 = 19

Example 2 2 5 5 6 7 10 15 21 21 23 23 25

n = 12,then ¼ n + ½ = ¼ (12)+½ = 3.5 then the Q1 in position 3 & 4 which is (5+6)/2=5.5

Q2 in position 9 & 10 which is (21+23)/2=22

Percentiles

Just as there are three quartiles separating data into four parts, there are 99 percentiles denoted P1, P2, . . . P99, which partition the data into 100 groups.

Percentile of value x = • 100number of values less than x

total number of values

n total number of values in the data set

k percentile being used

Notation

Converting from the kth Percentile to the Corresponding Data Value

nK

100

P oflocation k

Find the percentile corresponding the weight of 0.8143& find P10, P25

Example 1

81465.02

0.8150.8143 10&9 936

100

25

0.8073 4 6.336100

10

2210036

80.8143 of percentil

thth25

th10

22

betweenP

P

P

Solution

Interquartile Range (or IQR): Q3 - Q1

10 - 90 Percentile Range: P90 - P10

Semi-interquartile Range:2

Q3 - Q1

Midquartile:2

Q3 + Q1

Some Other Statistics

Recap

In this section we have discussed:

z Scores

z Scores and unusual values

Quartiles

Percentiles

Other statistics

Section 3-5Exploratory Data Analysis (EDA)

This section discusses outliers, then introduces a new statistical graph called a boxplot, which is helpful for visualizing the distribution of data.

Key Concept

Important Principles

An outlier can have a dramatic effect on the mean.

An outlier can have a dramatic effect on the standard deviation.

An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.

For a set of data, the 5-number summary consists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value.

A boxplot is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.

Definitions

Boxplots

Boxplots – cont.

Boxplots – cont.

Boxplots – cont.

Boxplots - Example

Recap

In this section we have looked at:

Exploratory Data Analysis

Effects of outliers

5-number summary

Boxplots

General Examples

Example 1

1612

192

n

xx

Fine mean, median, mode, midrange

x27817111525161414141318

192

x81113141414151617182527

5.172

278

14

5.142

1514

Midrange

Mode

Median

Solution

Example 2

0)5(6

9144.4)025.4(6

)1(

2

22

s

nn

xxns

Fine Standard deviation, variance for each of the two sample

x x2

0.8192 0.67110.815 0.66420.8163 0.66630.8211 0.67420.8181 0.66930.8247 0.68014.9144 4.025

Coke

x x2

0.8258 0.68190.8156 0.66520.8211 0.67420.817 0.66750.8216 0.6750.8302 0.68924.9313 4.053

Pepsi

32

22

1006.3)5(6

9313.4)053.4(6

)1(

s

nn

xxns

Example 3

xz

62.0

20.989.2

62.0

2.98100

100 )

z

xa

262.0

2.9896.96

96.96 )

z

xb

062.0

2.982.98

2.98 )

z

xc

Example 4Fine the indicated quartile or percentilea) Q1, b) Q3, c) P80, d) P33

Q1 position = ¼ n + ½ = ¼ (36)+½ = 9.5 (between 9th – 10th)

Q1= ( 0.8143+0.815 )/2=0.8147

Q3= ( 0.8207+0.8211 )/2=0.8209

8229.0 12 88.1136100

33

0.8152 92 8.2836100

80

th25

th80

P

P

Example 5Draw the boxplot for the following data set

-1 0 0 0 0 0 0 0 0 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 4 4 4 4 5 5 5 5 5 6 6 7 13 Sum = 139

673.252

139

n

xx

Solution

13 valuemaximum

1- valueminimum

5.32

43

214&13

5.1352 oflocation

2

22

22

3

114

21

41

21

41

1

Q

Qbetween

nQ

Mode

Median

th

Flash points

Which measure of center is the only one that can be used with data at the nominal level of measurement?

A. Mean

B. Median

C. Mode

Which of the following measures of center is not affected by outliers?

A. Mean

B. Median

C. Mode

Find the mode (s) for the given sample data.

79, 25, 79, 13, 25, 29, 56, 79

A. 79

B. 48.1

C. 42.5

D. 25

Which is not true about the variance?

A. It is the square of the standard deviation.

B. It is a measure of the spread of data.

C. The units of the variance are different from the units of the original data set.

D. It is not affected by outliers.

Weekly sales for a company are $10,000 with a standard deviation of $450. Sales for the past week were $9050. This is

A. Unusually high.

B. Unusually low.

C. About right.

In a data set with a range of 55.1 to 102.8 and 300 observations, there are 207 data points with values less than 88.6. Find the percentile for 88.6.

A. 32

B. 116.03

C. 69

D. 670

H.W 2Fine mean, median, mode, midrange, range, standard deviation, variance, P30Then draw the Boxplot

Age of US President