I Descriptive Statistics (PART I)

Introductory Statistics

ECON1005

I Descriptive Statistics (PART I)

■

Introduction

INTRODUCTION

■

What is Statistics?

■

Basic Definitions

What is Statistics?

•

Statistics

is a group of methods used to

Branches of Statistics

STATISTICS

DESCRIPTIVE STATISTICS

(characterise attributes of sample

&/or population)

•

Used to collect, organize, display and analyse data

•

There are two types:

1.

Numerical

●

Involves the computation of a statistic (eg. the average)

2.

Graphical

●

Involves representing the data using pictures

Inferential Statistics

•

Uses sample results to make generalizations,

inferences and predictions about a wider population

•

There are two types:

▫ Estimation

●

sample is used to estimate a parameter

▫ Hypothesis Testing

Descriptive vs. Inferential Statistics

•

Descriptive Statistics

▫ Collect

▫ Organize

▫ Summarize

▫ Display

▫ Analyze

•

Inferential Statistics

▫ Predict and forecast

values of a population

▫ Test hypotheses about

values of a population

Basic Definitions I

•

Population

:

▫ A

population

is the collection of all items whose characteristics

are being studied.

●

N represents the population size

▫ Values calculated using population data are called parameters

•

Sample:

▫ A

sample

is a portion of the population selected for study.

●

n represents the sample size

Basic Definitions II

•

Data:

▫ numbers or measurements that are collected

•

Variables:

▫ characteristics or attributes that enable us to distinguish one

individual from another

▫ they take on different values when different individuals are

observed (e.g. height)

•

Element:

Summarising & Describing Data

•

Describing the observed patterns in data is an

important part of statistics

•

Distribution of a single variable

Describing Data

S

Shape

What is the overall shape of the distribution?

_{(symmetric or skewed / Mounded or flat)}

U

Unusual

_{(errors, outliers or influential points)}

Are there any unusual points?

M

Middle

Where is the centre of the distribution?

_{(mean, median, mode)}

Organizing and Graphing Data

■

Introduction

■

Frequency Distributions

■

Bar Charts

Introduction

•

There are two main types of data:

▫

Quantitative

●

This is information presented in the form of numbers,

percentages or statistics

●

It answers in numerical terms such questions as "how often"

and "how many“

▫

Qualitative

Organizing Data

A frequency distribution lists all categories/classes and

the number of elements that belong to each

Frequency Distributions

•

A frequency distribution:

▫ a table in which measurements are tallied

▫ then the frequency or total number of times that each item

occurs is recorded

•

Usually measurements are arranged in ascending or descending

order

•

A frequency distribution has 3 columns

▫ the data categories or classes

▫ the tally column (for raw data)

▫ the corresponding frequencies

Examples

•

Quantitative

•

Qualitative

CATEGORY TALLY FREQUENCY

Yes 23

No 13

Undecided 4

Total 40

10M - 14M 25

15M - 19M 15

20M - 24M 19

25M - 29M 8

Total 77

Frequency Distribution Cont’d

•

two main types of frequency distributions:

▫ Ungrouped data

▫ Grouped data

Ad. expenditure (J$M) Tally Number of Firms (Frequency)

5M - 9M 10

10M - 14M 25

15M - 19M 15

20M - 24M 19

25M - 29M 8

Total 77

Ages Tally Frequency

•

Class Intervals/Limits

▫ largest or smallest numbers which can actually

belong to each class

▫ each class has a lower class limit and an upper

class limit

Ad. expenditure

(J$M)

Tally Number of Firms

5 - 9 10

10 – 14 25

15 – 19 15

20 – 24 19

25 - 29 8

▫ the numbers which separate classes

▫ given by the midpoint of the upper limit of one class and the

lower limit of the next class

Ad. expenditure (J$M) Class Boundaries Tally Number of Firms

5 - 9 10

10 – 14 9.5 – 14.5 25

15 – 19 15

20 – 24 19

25 - 29 8

Total 77

Lower class boundary for 2

class

(10 – 14):

•

Class Mark (Midpoint)

▫ found by taking the average of the class limits (or class

boundaries)

5 - 9 4.5 – 9.5 7 10

10 – 14 9.5 – 14.5 25

15 – 19 14.5 – 19.5 15

20 – 24 19.5 – 24.5 19

25 - 29 24.5 – 29.5 8

Total 77

Class 1 - Using Class Limits

•

Class Width

▫ aka: class size, class width, class length

▫ Two ways of calculating

▫ Method 1: the difference between corresponding class limits

▫ Method 2: the difference between two class boundaries

Ad. expenditure (J$M) Class Boundaries Midpoints Number of Firms

5 - 9 4.5 – 9.5 7 10

10 – 14 9.5 – 14.5 12 25

15 – 19 14.5 – 19.5 17 15

20 – 24 19.5 – 24.5 22 19

25 - 29 24.5 – 29.5 27 8

Total 77

Using Class

Limits

•

Found by dividing the frequency of a category/class by the

sum of all frequencies

▫ The sum of the relative frequencies MUST add to 1

▫ Sometimes expressed as a percentage

Ad. expenditure (J$M) Class Boundaries Number of Firms Relative Frequency

5 - 9 4.5 – 9.5 10 0.13

10 – 14 9.5 – 14.5 25

15 – 19 14.5 – 19.5 15

20 – 24 19.5 – 24.5 19

25 - 29 24.5 – 29.5 8

Total 77 1.00

General

Formula

Distributions

1.

The classes must be “mutually exclusive” - no element can

belong to more than one class

2.

Even if the frequency is zero, include each and every class

3.

Make all classes the same width (open ended classes may be

inevitable)

4.

Target between 5 and 20 classes, depending on the range and

number of data points

Consider the following data set:

2.3 4.2

2.8 6.7 4.7 1.6

2.0 1.4 1.0

2.8 1.8

5.2 6.0 5.2 3.5 1.0 3.6 5.1

1.9 7.3

2.5 5.6 3.3 3.4

2.9 3.0 1.8

2.1 3.1

2.8 2.1 4.3 7.1

4.9 1.6

2.2

4.5 6.3

2.7 8.3

a.

Group these figures into a frequency distribution having the classes:

1.0 – 1.9, 2.0 – 2.9, 3.0 – 3.9, 4.0 – 4.9, 5.0 – 5.9, 6.0 – 6.9, 7.0 –

7.9, and 8.0 – 8.9

b.

Calculate the class boundaries

c.

Calculate the class midpoints

d.

Calculate the class width

Graphical Representation

•

When presenting

Quantitative Data

use:

▫ histograms

▫ frequency polygons

▫ cumulative frequency polygons (O-give)

•

When presenting

Qualitative Data

use:

▫ A graphical way of presenting qualitative data

•

Bars (columns) are separated from each other and have

the same width

•

Categories are placed on the horizontal axis and

▫ A graphical way of presenting qualitative data

•

Pie Chart is a circle divided into portions that represent

the relative frequencies belonging to different categories.

•

To construct pie chart:

Qualitative Example

The following are the results for a third year

statistics course:

A - 41

B+ - 12

B - 2

C - 22

F - 19

▫ Calculate the relative frequencies

▫ Construct a bar chart

•

A graphical way of presenting qualitative data

•

Divide data into classes of equal width and the number of

observations in each class is counted (information would be

presented in a frequency table)

•

Class is on the x-axis (horizontal)

▫ Can plot using either:

●

Class Limits

●

Class Boundaries

•

Frequency (or relative frequency) is on the y-axis (vertical)

•

Bars are drawn where the base of each bar covers the class

and the height of each bar covers the frequency

Figure 2 – plotted using class

limits

4. Frequency Polygons

•

A

Frequency Polygon

is a line graph joining the

midpoints of the bars of a histogram

•

To construct a frequency polygon:

▫ Plot the midpoint of each class (on horizontal) with its

corresponding frequency/relative frequency (on vertical)

•

Examines how many observations

lie below

a certain class

boundary

•

Plotted against the upper class boundaries

Using Frequencies

•

The first value in the distribution is ALWAYS zero

•

The last value in the distribution is ALWAYS the total number

Using Relative Frequencies

•

The first value in the distribution is ALWAYS zero

•

The last value in the distribution is ALWAYS 1

Using Percentages

•

The first value in the distribution is ALWAYS zero

expenditure (J$M)

Upper Class Boundaries

Number

of Firms Cumulative Frequency

4.5

5 - 9 9.5 10

10 – 14 14.5 25

15 – 19 19.5 15

20 – 24 24.5 19

25 - 29 29.5 8

Total - 77

•

Examines how many observations

lie above

a certain class

boundary

•

Plotted against the upper class boundaries

Using Frequencies

•

The first value in the distribution is ALWAYS the total

number

•

The last value in the distribution is ALWAYS zero

Using Relative Frequencies

•

The first value in the distribution is ALWAYS 1

•

The last value in the distribution is ALWAYS zero

Using Percentages

•

The first value in the distribution is ALWAYS 100

Ad. expenditure (J$M) Upper Class Boundaries Number of Firms More Than Cumulative Frequency 4.5

5 - 9 9.5 10

10 – 14 14.5 25

15 – 19 19.5 15

20 – 24 24.5 19

25 - 29 29.5 8

Total - 77

•

Using the example on

slide 25

,

a.

Construct a histogram with a superimposed

frequency polygon

b.

Calculate:

●

Less than cumulative frequencies

●

More than cumulative frequencies

c.

Construct the:

●

Less than cumulative frequency ogive