Chapter 3Descriptive Measures

Measures

Central Tendency

Mean Median Mode

Dispersion

RangeVariance &Standard Deviation

Measures

Central Tendency

Mean Median Mode

Dispersion

RangeVariance &Standard Deviation

The 3-Ms: Mean, Median, ModeMean: arithmetic averageMedian: mid-point of the distributionMode: most frequent responseWhich one to use?Depends on the type of data you have:Nominal data: modeOrdinal data: mode and medianInterval/Ratio: mode, median, and mean

Mean (Arithmetic)—the common measure of CT affected by extreme values(outliers)

Sample mean

n is sample size.Population mean

N is population size

Example: All the students in IT A3-04 class are

considered the population. Their course grades are:86 92 70 65 55 89 90 95 4985

(i). Give the formula for the population mean(ii). Compute the mean course grade

Solution:(i). See the previous slide(ii). Compute the population mean µ

µ=(86+92 +70 +65+55+89+90+95+49+85)/10 =77.6

Median Mean is affected by extreme values. So, it can not

represent a data set that contains one or two very large or very small values. The center point for such data can be better described using Median.

In an ordered array, the median is the middle valueThe location of the media is (n+1)/2 If the number of values is odd the median is the

middle number If the number of values is even the median is the

average of the two middle numbers.

Example:The prices ordered at CP are:280 300 180 320 700Calculate the median of the data set above.

Solution:-Ordered array: 180 280 300 320 700-Median position=(n+1)/2

=(5+1)/2=3th=>Median=300

ModeA measure of CTValue that occurs most oftenNot affected by extreme valuesThere may not be a modeThere may be several modesUser for numerical or categorical data

Example:The following are the ages of the 9 people in CP:19 20 27 25 30 32 40 50

20What is the mode of the data set?

Solution:The data set reveals that the age 20 appear most often than any other ages. So the mode is 20.

Why measures Dispersion? It tells us how often something varies/spread. Imagine you go to a restaurant and received such

very good mean, you return fore the same meal next day.

However, this time the food tastes very bad. You never go back. The standard of quality has varied.

The mean or the median only locates the center of the data. But they do not inform us about the spread of the data.

A measure of dispersion can be used to evaluate the reliability of two or more means/averages.

RangeThe simplest measure of dispersion is the

range.It is the difference between the lowest and

the highest.It can be expressed by the following formula:

Range=Highest value-Lowest valueExample:In the morning IT class, students’ grades are 45 45 50 52 65 75 82 82 90In the evening IT class, students’ grades are40 45 45 50 50 50 60 65 70 70

IT morning class:Range=90-45=45

Mean=(45+45+50+52+65+75+82+82+90)/9 =65.11

IT evening class:Range=70-40=30 Mean=(40+45+45+50+50+50+60+65+70+70)/10

=55Therefore, the range of students’ grades of IT morning class is greater than the range of students’ IT evening class. This we can conclude that there is greater dispersion in students’ grades of IT morning class than in the students’ grades of IT evening class. And we also conclude that the students grades of IT morning class

Are not clustered more closely around the mean of 65.11 than the students’ grades of IT evening class. Thus, the mean of 55 are more reliable than the mean of 65.11.

Variance and Standard DeviationThe defect of the range is that it is based only

on two values—the highest value and the lowest value.

Variance is the arithmetic mean of squared deviation from the mean.

Variance is the important measure of variation.

Standard Deviation is the positive square root of the variance.

Standard Deviation is the most important measure of variation.

Population variance:

Population Standard Deviation

Sample variance

Sample Standard Deviation

Small vs. large standard deviation

small standard deviation

Large standard deviation

Example: Computer the population variances and

standard deviation of the previous students’ grades. And then make a conclusion based on the two standard deviations.

Exercises1. Based on a given data set (sample):

5, 4, 2, 7, 4, 8, and 2a. Compute the varianceb. Determine the sample standard deviation

2. Listed below are self-services prices for a sample of 16 retail stores:1.25 1.19 1.29 1.21 1.27 1.21 1.26 1.271.23 1.25 1.23 1.21 1.24 1.23 1.25 1.24a. What is the arithmetic mean selling price?b. What is the median selling price?c. What is the modal selling price?

3. Listed below are he numbers of boxes of cigarettes produced daily in Luxury cigarette manufacture.1000 1200 1000 1100 1500 1450 12001100 1300 1300 1400 1300 1200 13001300 1200 1300 1100 1400 1300 12001400 1300a. Find the mean, median, and mode of the data setb. Calculate the standard deviationc. Draw polygon to present the data set and provide the comment about the distribution of the data set.