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

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

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 Madgerie Jameson 5/18/2019

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

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. Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

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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. Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

Student Rank 1 2 3 4 John Maths Language Arts Science Music Mary Susan Madgerie Jameson 5/18/2019

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”. Madgerie Jameson 5/18/2019

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 1 2 3 4 Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

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 200. 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. Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

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. Madgerie Jameson 5/18/2019

Summary Scale of Measurement Nominal Ordinal Interval Ratio Properties Identity Magnitude Equal interval True Zero Mathematical Operations Count Rank order Addition Subtraction Multiplication Division Madgerie Jameson 5/18/2019

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 Madgerie Jameson 5/18/2019

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

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

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

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

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: 58 - 12 = 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

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 10 - 19 3 .15 15 20 but less than 30 20 – 29 6 .30 30 30 but less than 40 30 - 39 5 .25 25 40 but less than 50 40 - 49 4 .20 20 50 but less than 60 50 – 59 2 .10 10 Total 1.00 100

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

Histogram Example (No gaps between bars) Class Midpoints Frequency 10 but less than 20 15 3 20 but less than 30 25 6 30 but less than 40 35 5 40 but less than 50 45 4 50 but less than 60 55 2 (No gaps between bars) Class Midpoints

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

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)

Time to Practice Here is a frequency distribution. Create a histogram using the data. Class interval frequency 900 - 100 12 80 - 89 14 70 - 70 20 60 – 60 24 50 – 59 28 40 – 49 30 -39 21 20 – 29 15 10 – 19 17 0 – 9 Madgerie Jameson 5/18/2019

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