EDU 660

Methods of Educational ResearchDescriptive Statistics

John Wilson Ph.D.

DefinitionsQuantitative data

numbers representing counts or measurements

DefinitionsQuantitative data

numbers representing counts or measurements

Qualitative (or categorical or attribute) data

can be separated into different categories that are distinguished by some non-numeric characteristics

DefinitionsQuantitative data

the incomes of college graduates

DefinitionsQuantitative data

the incomes of college graduates

Qualitative (or categorical or attribute) data

the genders (male/female) of college graduates

Discrete data result when the number of possible values is

a ‘countable’ number0, 1, 2, 3, . . .

Definitions

Discrete data result when the number is or a ‘countable’

number of possible values0, 1, 2, 3, . . .

Continuous (numerical) data result from infinitely many possible

values that correspond to some continuous scale

Definitions

2 3

Discrete

The number of students in a classroom.

Definitions

Discrete

The number of students in a classroom.

ContinuousThe value of all coins carried by the students in the classroom.

Definitions

nominal level of measurement characterized by data that consist of names, labels, or categories only. The data cannot be arranged in an ordering scheme (such as low to high)

Example: Your car rental is a: Ford, Nissan, Honda, or Chevrolet

Levels of Measurement of Data

ordinal level of measurement involves data that may be arranged in some order, but differences between data values either cannot be determined or are meaningless.

Example: Course grades A, B, C, D, or F. Your car rental is an: economy, compact, mid-size, or full-size car.

Levels of Measurement of Data

interval level of measurement like the ordinal level, with the additional property that the

difference between any two data values is the same. However, there is no natural zero starting point (where

none of the quantity is present)

Example: The temperature outside is 5 degrees Celsius.

Levels of Measurement of Data

ratio level of measurementthe interval level modified to include the natural zero

starting point (where zero indicates that none of the quantity is present). For values at this level, differences and ratios are meaningful.

Examples: Prices of textbooks.

The Temperature outside is 278 degrees Kelvin.

Levels of Measurement of Data

Levels of Measurement Nominal - categories only

Ordinal - categories with some order

Interval – interval are the same, but no natural starting point

Ratio – intervals are the same, and a natural starting point

a value at the centre or middle of a data set

MeanMedianMode

Measures of the centre

Mean(Arithmetic Mean)

AVERAGEThe number obtained by adding the values and dividing the total by the number of

values

Definitions

Notation denotes the addition of a set of values

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

n represents the number of data values in a sample

N represents the number of data values in a population

Notationis pronounced ‘x-bar’ and denotes the mean of a set of sample values

x =n x

x

Notation

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

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

x =n x

x

Nµ =

x

Definitions

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

The Median is used to describe house prices in Toronto. Why not the Mean?

Definitions Mode

the score that occurs most frequentlyBimodal

MultimodalNo Mode

denoted by M

the only measure of central tendency that can be used with nominal data

a. 5 5 5 3 1 5 1 4 3 5

b. 1 2 2 2 3 4 5 6 6 6 7 9

c. 1 2 3 6 7 8 9 10

Examples

Mode is 5

Bimodal - 2 and 6

No Mode

Waiting Times of Bank Customers at Different Banks(in minutes)

TD

RBC

6.5

4.2

6.6

5.4

6.7

5.8

6.8

6.2

7.1

6.7

7.3

7.7

7.4

7.7

7.7

8.5

7.7

9.3

7.7

10.0

TD

RBC

6.5

4.2

6.6

5.4

6.7

5.8

6.8

6.2

7.1

6.7

7.3

7.7

7.4

7.7

7.7

8.5

7.7

9.3

7.7

10.0

TD

7.15

7.20

7.7

7.10

RBC

7.15

7.20

7.7

7.10

Mean

Median

Mode

Midrange

Waiting Times of Bank Customers at Different Banksin minutes

Measures of Variation

RangeVarianceStandard Deviation

Measures of Variation

Range

valuehighest lowest

value

Measures of Variation

Variance

• Mean Squared Deviation from the Mean

(Root Mean Squared Deviation)

Measures of Variation

Standard Deviation

Population Standard Deviation Formula

Root Mean Squared Deviation

(x - x)2

N s=

Basketball Starting LineHeight (inches)

78

77

75

74

71