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Chapter 1

Organization and description of data

1.1. Introduction

Statistics is a group of methods that are used to collect, analyse, present, interpret data and make decisions.

Statistics is sometimes divided into two main areas:

1. Descriptive statistics

2. Inferential statistics.

Descriptive statistics consists of the collection, organization, summation, and presentation of data.

A population is a complete set of units (usually people, objects, events) that we are interested in studying.

A subset of the population selected for study is called a sample.

Inferential statistics is an estimate or prediction about a population based on information contained in a sample.

1.2. The mean

The mean for ungrouped data, also known as the arithmetic average, is found by adding the values of the data and dividing by the total number of values. Thus,

Mean for population data:

Mean for sample data:

where is the population size, is sample size, (Greek letter mu) is the population mean, and (read as “-bar ”) is the sample mean.

Example:

Calculate the mean of the following six sample observations:

5, 2, 6, 8, 7, 8

Solution:

Using the definition of sample mean, we find

.

Thus, the mean of this sample is 6.

Example:

The salaries of all 7 employees of a small company are:

$ 320, 410, 310, 480, 530, 370, 240

Find the mean salary.

Solution:

Since the given data set includes all 7 employees of the company, it represents the population. Hence, . The population mean is

.

Thus, the mean salary of the employees of this company is $380.

1.3. The median

The median is the middle term in a data set. Before one can find this point the data must be arranged in increasing (or decreasing) order. The calculation of the median for ungrouped data consists of the following two steps:

1. Rank the given data set in increasing (or decreasing) order.

2. Find term in a ranked data set.

The value of term is the median.

There are two possibilities

1) If is odd, then the median is given by the value of the middle term in a ranked data.

2) If is even, then the median is given by the average of the values of the two middle term.

Remark: If the given data set represents a population, replace by .

Example:

Consider again the seven salaries of employees of a small company

$ 320, 410, 310, 480, 530, 370, 240

Calculate the median of this population.

Solution:

First of all, let us rank salaries in ascending order:

$ 240, 310, 320, 370, 410, 480, 530

N=7 and

Therefore, the median is the value of the fourth term in the ranked data

$ 240, 310, 320, 370 , 410, 480, 530

Thus, the median value for this population is $370.

Example:

The ages of a sample of 10 university students are

18, 22, 19, 20, 21, 18, 22, 19, 23, 17

Calculate the median of this sample.

Solution:

First we order the data in increasing order. The ordered vales are

17, 18, 18, 19, 19, 20, 21, 22, 22, 23

There are 10 values in the data set. Hence,

and

Therefore, the median is given by the mean of fifth and sixth values in the ranked data.

,

.

Hence, the median age is 19.5.

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