Download Descriptive Statistics

EDU 660
Methods of Educational Research
Descriptive Statistics
John Wilson Ph.D.
Definitions
Quantitative data
numbers representing counts or
measurements
Definitions
Quantitative 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
Definitions
Quantitative data
the incomes of college graduates
Definitions
Quantitative data
the incomes of college graduates
 Qualitative (or categorical or
attribute) data
the genders (male/female) of college
graduates
Definitions
Discrete
data result when the number of possible values is
a ‘countable’ number
0, 1, 2, 3, . . .
Discrete
Definitions
data result when the number is or a ‘countable’
number of possible values
0, 1, 2, 3, . . .
 Continuous
(numerical) data result from infinitely many
possible values that correspond to some continuous
scale
2
3
Definitions
Discrete
The number of students in a classroom.
Discrete
Definitions
The number of students in a classroom.
 Continuous
The value of all coins carried by the students in the classroom.
Levels of Measurement of Data
 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 measurement
the 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
 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
Measures of the centre
a value at the centre or middle of a data set
Mean
Median
Mode
Definitions
Mean
(Arithmetic Mean)
AVERAGE
The number obtained by adding the values and
dividing the total by the number of values
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
Notation
pronounced ‘x-bar’ and denotes the mean of a set of
x issample
values
x
x =
n
Notation
pronounced ‘x-bar’ and denotes the mean of a set of
x issample
values
x
x =
n
µ
is pronounced ‘mu’ and denotes the mean of all values in a population
µ =
x
N
Definitions

Median
the 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 frequently
Bimodal
Multimodal
No Mode
denoted by M
the only measure of central tendency that can be used with nominal data
Examples
a. 5 5 5 3 1 5 1 4 3 5
Mode is 5
b. 1 2 2 2 3 4 5 6 6 6 7 9
Bimodal -
c. 1 2 3 6 7 8 9 10
No Mode
2 and 6
Waiting Times of Bank Customers
at Different Banks
(in minutes)
TD
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
7.7
7.7
RBC
4.2
5.4
5.8
6.2
6.7
7.7
7.7
8.5
9.3
10.0
Waiting Times of Bank Customers
at Different Banks
in minutes
TD
6.5
6.6
6.7
6.8
7.1
7.3
7.4
7.7
7.7
7.7
RBC
4.2
5.4
5.8
6.2
6.7
7.7
7.7
8.5
9.3
10.0
RBC
TD
Mean
7.15
7.15
Median
7.20
7.20
Mode
7.7
7.7
Midrange
7.10
7.10
Measures of Variation
Range
Variance
Standard Deviation
Measures of Variation
Range
highest
value
lowest
value
Measures of Variation
Variance
• Mean Squared Deviation from the Mean
Measures of Variation
Standard Deviation
(Root Mean Squared Deviation)
Population Standard Deviation Formula
s=
 (x - x)
N
2
Root Mean Squared Deviation
Basketball Starting Line
Height (inches)
78
77
75
74
71