IT Colleges Introduction to Statistical Computer Packages Lecture 3 Eng. Heba Hamad week

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 Concept When 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 Introduction to Statistics

Population Mean Example There 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 No. of PatentsCompanyNo. of PatentsCompany 210Mazda511General Motors 97Chrysler385Nissan 50Porsche275DaimlerChrysler 36Mitsubishi257Toyota 23Volvo249Honda 13BMW234Ford 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 values 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 What is the arithmetic mean number of minutes used? Sample mean = 97.5 minutes 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

(in order - even number of values – no exact middle shared by two numbers) MEDIAN is Introduction to Statistics

(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 Multimodal No Mode Introduction to Statistics

a b c 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 I Data Set II For each data set determine: Mean Median Mode Introduction to Statistics

Example 1 Solution Introduction to Statistics

The weighted mean The 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* *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. x Sum Introduction to Statistics

Best Measure of Center Introduction to Statistics

 Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half  Skewed distribution 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