1. The University of the West Indies School of Education Introduction to Statistics
Lecture two
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5/18/2019

2. Topics to be covered in this lecture
Scales of Measurement
Frequency distribution
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5/18/2019

3. Scales of Measurement
In statistics measurement is categorised according to levels. Each level corresponds to how the measurement can be treated mathematically. The assignment of numbers determines the scale of measurement. Four scales of measurement are typically used in Education statistics
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5/18/2019

4. Contd. Nominal Ordinal Interval Ratio
Each scale represents a particular property of set of properties of the abstract number system.
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5/18/2019

5. Properties of the abstract number system
The properties of the number system that are
relevant to the scale of measurement are
Identity the number has a particular meaning.
Magnitude the numbers have an inherent order from smaller to larger
Equal intervals the differences between numbers (units) anywhere on the scale are the same.
Absolute/true zero the zero point represents the absence of the property being measured.
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6. Nominal Scale The lowest scales of measurement
Numbers are assigned to categories as “names”
The number gives the identity of the category assigned.
The only mathematical operation that can be performed with nominal scale is to count.
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5/18/2019

7. Example
Classifying people according to sex is a common nominal scale. In the following example 1 is assigned to male, 2 is assigned to female.
Notice when the number assigned to male and female changes there is no impact on the data representation.
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8. Madgerie Jameson
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9. Ordinal Scales
Ordinal scales have the property of identity and magnitude.
The numbers represent a quality being measured (identity)
It can tell whether a case has more or less than the quality measured (magnitude).
The distance between the scale points is not equal.
Ranked preferences are presented as an example of ordinal scales.
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5/18/2019

10. Example
As educators we are interested in students preferences for different subjects. The subject choice is usually done for CXC examinations. Let’s say we asked the three students below to rank their preferences for four different subjects. We usually rank our strongest preference with 1. With the four subjects the lowest preference would be 4. For the four subjects we assign a rank which tells us the order (magnitude) of the preference for the particular subject (identity). The number tells us that the students prefer one subject over the other but how much they prefer the subject.
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11. Student Rank 1 2 3 4 John Maths Language Arts Science Music Mary
Susan
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12. Distance between the scale
We assume that the intervals between scale points on ordinal scales are unequal
Therefore the distance between a rank of “1” and “2” is not necessarily the same as the distance between the ranks of “3” and “4”.
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5/18/2019

13. Maths Lang Arts Science Music
Let us say that our first student likes Maths the best but also has a strong liking for Language Arts. He thinks Science is ok and he really does not like Music. In that case the preference distance between “3” and “4” is much greater than the preference distance between ranks “1” and “2”.
Maths Lang Arts Science Music
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14. Interval Scales
Interval scales have the properties of identity, magnitude and equal distance.
The equal distance between the scale points allows us to identify the number of units greater than or less than one case is from another on a measured characteristic.
We are confident that the meaning of the distance between 15 and 20 is the same as the distance between 85 and 90.
Interval scales do not have a true zero point.
The number “0” is arbitrary.
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5/18/2019

15. Example
An example of an interval scale is the measurement of temperature on Fahrenheit or Celsius scales.
We know that 600 is hotter than 300 and there is a 5 degree difference in temperature between 150 and Zero degrees on either scale is an arbitrary number and not a “ true Zero”. The zero point does not indicate an absence of temperature; it is an arbitrary point on a scale.
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16. Ratio Scales
Ratio scales of measurement have all the properties of the abstract number, identity, magnitude, equal distance, and absolute zero.
The properties allow us to apply all the possible mathematical operations ( addition, subtraction, multiplication and division) in data analysis.
The absolute zero allows us to know how may times greater one case is than the other.
Scales with an absolute zero and equal intervals are considered ratio scales.
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5/18/2019

17. Example
Money is a good example of a ratio scale of measurement. If we have $1000 we have twice as much purchasing power as $500. If we have no money, we have absolutely no ability to purchase anything.
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18. Summary Scale of Measurement Nominal Ordinal Interval Ratio Properties
Identity
Magnitude
Equal interval
True Zero
Mathematical Operations
Count
Rank order
Addition
Subtraction
Multiplication
Division
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19. Statistical Applications
Nominal
Ordinal
Interval
Ratio
Descriptive Statistics
Mode
Median
Range statistics
Mean
Variance
Standard
deviation
Standard deviation
Inferential Statistics
Non Parametric
Chi-square
Non parametric
ANOVA
Spearman Correlation
Parametric
T test
Pearson Correlation
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5/18/2019

20. Tabulating Numerical Data: Frequency Distributions
What is a Frequency Distribution?
A frequency distribution is a list or a table …
containing class groupings (categories or ranges within which the data fall) ...
and the corresponding frequencies with which data fall within each grouping or category

21. Why Use Frequency Distributions?
A frequency distribution is a way to summarize data
The distribution condenses the raw data into a more useful form...
and allows for a quick visual interpretation of the data

22. Class Intervals and Class Boundaries
Each class grouping has the same width
Determine the width of each interval by
Use at least 5 but no more than 15 groupings
Class boundaries never overlap
Round up the interval width to get desirable endpoints

23. Frequency Distribution Example
Example: The ministry of education randomly selects 20 schools and records the average test score of each:
24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
32, 13, 12, 38, 41, 43, 44, 27, 53, 27

24. Frequency Distribution Example
(continued)
Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Find range: = 46
Select number of classes: 5 (usually between 5 and 15)
Compute class interval (width): 10 (46/5 then round up)
Determine class boundaries (limits): 10, 20, 30, 40, 50, 60
Compute class midpoints: 15, 25, 35, 45, 55
Count observations & assign to classes

25. Frequency Distribution Example
(continued)
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class
Frequency
Relative Frequency
Percentage
10 but less than20 3
.15 15
20 but less than – 29 6
.30 30
30 but less than 5
.25 25
40 but less than 4
.20 20
50 but less than – 59 2
.10 10
Total
1.00
100

26. Graphing Numerical Data: The Histogram
A graph of the data in a frequency distribution is called a histogram
The class boundaries (or class midpoints) are shown on the horizontal axis
the vertical axis is either frequency, relative frequency, or percentage
Bars of the appropriate heights are used to represent the number of observations within each class

27. Histogram Example (No gaps between bars) Class Midpoints
Frequency
10 but less than
20 but less than
30 but less than
40 but less than
50 but less than
(No gaps between bars)
Class Midpoints

28. Questions for Grouping Data into Classes
1. How wide should each interval be? (How many classes should be used?)
2. How should the endpoints of the intervals be determined?
Often answered by trial and error, subject to user judgment
The goal is to create a distribution that is neither too "jagged" nor too "blocky”
Goal is to appropriately show the pattern of variation in the data

29. How Many Class Intervals?
Many (Narrow class intervals)
may yield a very jagged distribution with gaps from empty classes
Can give a poor indication of how frequency varies across classes
Few (Wide class intervals)
may compress variation too much and yield a blocky distribution
can obscure important patterns of variation.
(X axis labels are upper class endpoints)

30. Time to Practice
Here is a frequency distribution. Create a histogram using the data.
Class interval
frequency 12 14 20
60 – 60 24
50 – 59 28
40 – 49
30 -39 21
20 – 29 15
10 – 19 17
0 – 9
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31. Identify the following examples below as representing the nominal, ordinal or ratio scales of measurement
Time.
Skin temperature in degrees centigrade.
The number of statistics questions answered correctly on a test
Class rank.
The position of a chicken in the current pecking order.
The name of the EEG tracing exhibited by a sleeping patient ( alpha, beta, theta, delta…).
The number of errors a student makes when reading a passage.
The number of errors on a colour vision test.
The total number of words recalled from a list of 50 words.
The number of stressful events you have experienced in the last six months.
Your rankings of the best 10 movies of last year.
The number of trials necessary for a child to learn how to tie his shoe laces.
The number of students in the class who are doing, science, youth guidance, curriculum, Mphil, PhD.
Madgerie Jameson
5/18/2019