IT Colleges

Introduction to Statistical Computer Packages

Lecture 3

Eng. Heba Hamad

week 3 - 2008

Chapter 2 (part 2)

Statistics for Describing Data

Introduction to Statistics

Statistics for Describing, Exploring, and Comparing Data

Measures of Center

Measures of Variation

Introduction to Statistics

Measures of Center

Introduction to Statistics

Key ConceptWhen describing, exploring, and comparing data sets, these characteristics are usually extremely important:

center,

variation,

distribution,

Outliers.

Introduction to Statistics

Definition

Measure of Center (Measures of location)

The value at the center or middle of a data set.

Measures of centers are often referred to as averages. The purpose of a measure of center is to pinpoint the center of a set of values.

Introduction to Statistics

Definition

Arithmetic Mean (Mean)

The measure of center obtained by adding the values and dividing the total by the number of values

Introduction to Statistics

Notation

denotes the sum of a set of values.

x is the variable usually used to represent the individual data values.

n represents the number of values in a sample.

N represents the number of values in a population.

Introduction to Statistics

Population Mean

The population mean is the sum of all values in the population divided by the number of the values in the population.

is pronounced ‘mu’ and denotes the mean of all values in a population

N

x

Introduction to Statistics

Population Mean ExampleThere are 12 automobile manufacturing companies in the United States. Listed below is the no. of patents granted by U.S. government to each company in a recent year

CompanyNo. of PatentsCompanyNo. of Patents

General Motors511Mazda210

Nissan385Chrysler97

DaimlerChrysler275Porsche50

Toyota257Mitsubishi36

Honda249Volvo23

Ford234BMW13

Introduction to Statistics

Population Mean Example

This is an example of a population mean because we are considering all the automobiles manufacturing companies obtaining patents.

=195

Introduction to Statistics

Sample Mean

The sample mean is the sum of all the sampled values divided by the total number of sampled values.

is pronounced ‘x-bar’ and denotes the mean of a set of sample valuesx

n

xx

Introduction to Statistics

Sample Mean Example

SunCom is studying the number of minutes used monthly by clients in a particular cell phone rate plan. A random sample of 12 clients showed the following number of minutes used last month

90 77 94 89 119 112 91 110 100 92 113 83

What is the arithmetic mean number of minutes used?

Sample mean = 97.5 minutesx

Introduction to Statistics

Properties of the Arithmetic Mean (Mean)

• All the values are included in the computing mean

• A set of data has only one mean. The mean is unique.

• The sum of the deviations of each value from the mean will be zero.

• The mean is affected by outliers

Introduction to Statistics

Exercise

There are 10 salespeople employed by midtown ford. The no. of new cars sold last month by respective salespeople were:

15,23,4,19,18,10,10,8,28,19

Compute the mean and indicate whether it is a statistic or a parameter.

Introduction to Statistics

Definitions

Median

the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

is not affected by an

extreme value

Introduction to Statistics

Finding the Median

If the number of values is odd, the median is the number located in the exact middle of the list.

If the number of values is even, the median is found by computing the mean of the two middle numbers.

Introduction to Statistics

5.40 1.10 0.42 0.73 0.48 1.10

0.42 0.48 0.73 1.10 1.10 5.40

2

(in order - even number of values – no exact middleshared by two numbers)

MEDIAN is 0.915

Introduction to Statistics

5.40 1.10 0.42 0.73 0.48 1.10 0.66

0.42 0.48 0.66 0.73 1.10 1.10 5.40

(in order - odd number of values)

exact middle MEDIAN is 0.73

Introduction to Statistics

Properties of the Median

• A set of data has only one median. The mean is unique.

• The mean is not affected by extremely large or smale values.

Introduction to Statistics

Definitions

Mode

the value that occurs most frequently

Mode is not always unique

A data set may be:Bimodal

MultimodalNo Mode

Introduction to Statistics

a. 5.40 1.10 0.42 0.73 0.48 1.10

b. 27 27 27 55 55 55 88 88 99

c. 1 2 3 6 7 8 9 10

Mode - Examples

Mode is 1.10

Bimodal - 27 & 55

No Mode

Introduction to Statistics

Midrange

The value midway between the maximum and minimum values in the original data set

Definition

Midrange = maximum value + minimum value

2

Introduction to Statistics

Example 1

Table 3.1

Table 3.2

Data Set IData Set II

For each data set determine: Mean Median Mode

Introduction to Statistics

Example 1

Solution

Introduction to Statistics

The weighted meanThe weighted mean is a special case of the arithmetic mean. It occurs when there are several observations of the same value. To explain, suppose we want to compute the mean for the following values: 9, 9, 9, 12.5, 12.5, 12.5, 12.5, 15, 15, 15 ,

Weighted mean = (3*9 +4*12.5 + 3*15) / 10

= 12.2

Introduction to Statistics

use class midpoint of classes for variable x

Mean from a Frequency Distribution

Class Mid point

Introduction to Statistics

Example 2

Class limitsClass Mid point xFrequency ff . x21 - 3025.52871431 - 4035.530106541 - 5045.51254651 - 6055.5211161 - 7065.5213171 - 8075.52151

Sum762718

76.3576

2718.

f

xfx

Introduction to Statistics

Best Measure of Center

Introduction to Statistics

Symmetricdistribution of data is symmetric if

the left half of its histogram is roughly a mirror image of its right half

Skeweddistribution of data is skewed if it is

not symmetric and if it extends more to one side than the other

Definitions

Introduction to Statistics

Skewness

Introduction to Statistics