# Nominal Scaling. Measures of Central Tendency, Spread, and Shape. Interval Scaling. Ordinal Scaling

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1 Nominal Scaling Measures of, Spread, and Shape Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning The lowest level of scaling In Nominal Scaling, each of the values serves as a representation. Each observation belongs to one mutually eclusive category and has no logical order Eamples: Gender Ethnicity School Ordinal Scaling Interval Scaling In Ordinal Scaling, each of the values is in rank order. Each observation belongs to one mutually eclusive category, but we now have logical order. Eamples: Letter Grades Place Finished in a Race Likert-type Scaling In Interval Scaling, each of the values has a specific order that reflects equal differences between observations. Each observation belongs to one mutually eclusive category, with logical order, and equal differences between each of the points and no absolute zero. Eamples: Temperature (Fahrenheit and Celcius) IQ Scores SAT/GRE Scores

2 Ratio Scaling Measures of In Ratio Scaling, each of the values has a specific order that reflects equal differences and a true zero. Each observation belongs to one mutually eclusive category, with logical order, equal differences between each of the points, and has a true zero. Eamples: Kelvin Scale Height and Weight Speed The measures of central tendency try and give us a picture of what is going on at the middle of the distribution There are 3 types of measures of central tendency The mode - this is the most frequently appearing score The median - this is also called the middle score The mean - this is also called the average score The mode is the most frequently appearing score There can be as many modes as there are pieces of data in a dataset Datasets with two modes are usually referred to as bimodal

3 Representing the Mode Suppose we have 2 separate datasets X = 1,2,2,3,3,3,4,4,5 Y = 1,1,1,2,2,3,4,4,4,5 Histogram of X Data Histogram of Y Data y The median is often called the middle score To compute the median, you first arrange the scores in ascending order For datasets with an odd number of scores, the median is the middle score. For datasets with an even number of scores, the median is the score halfway between the two middle scores If X = 1,2,3,4,1, the median would be 3 If X = 1,2,3,1, the median would be 2.5 If X = 3,2,1,1, the median would still be 2.5

4 The mean is also called the average score n n X = (X i )/n = (X i) n Given the dataset X = 1,2,3,4 X = = = 2.5 The range is used to describe the number of units on the scale of measurement It is computed as: (Highest score - Lowest score) Suppose we have the dataset X = 1,2,3,4, the range would be: (4-1) = 3

5 Computing the Suppose that we have a dataset where X = 1,2,3,4 The standard deviation is also sometimes referred to as the mean deviation The SD represents the average distance of each score from the mean n SD X = (X i X) 2 n 1 (1 2.5) SD X = 2 + (2 2.5) 2 + (3 2.5) 2 + (4 2.5) ( 1.5) = 2 + (.5) 2 + (.5) 2 + (1.5) = 3 5 = 3 = 1.67 = 1.29 As opposed to the other measure of variation, the z-score is a measure of deviation for a single individual, as opposed to a group of scores. The Z-score represents the number of units a given piece of data is from the mean. Z i = X i X SD X Raw Score Z-Score

6 The standard error of the mean is really just computed for the purposes of computing SEM X = SD X n For our dataset, X = 1,2,3,4, SEM X = =.645 Suppose that we have a uniform normal distribution of population scores between -1 and +1 [U( 1, 1)] and we take many samples of n= > mean(res) [1] e-5 dnorm(,, 1) % area at z= Area Under the Curve dnorm(,, 1) % area at z= > sd(res) [1]

7 Computing 95% Let s look back at our previous dataset where X=1,2,3,4. In this case, we noted that the mean of X = 2.5 and the SD of X = This would make the SEM = 1.29/2 = % confidence intervals for any dataset are computed as: CI 95% = X ± 1.96 SEM = 2.5 ± = 2.5 ± Graphing 95% > <- c(1, 2, 3, 4) > y <- c(1, 3, 5, 7, 9) > plotmeans(c(, y) ~ rep(c("", "y"), c(4, 5))) c(, y) n=4 n=5 y rep(c("", "y"), c(4, 5)) Thinking Through Suppose that we have two different datasets with the following properties: Thinking Through > study1 <- rnorm(3, 1, 5) > study2 <- rnorm(3, 1, 1) > plotmeans(c(study1, study2) ~ rep(c("study 1", + "Study 2"), c(3, 3))) Study 1 Study 2 mean 1 1 SD 5 1 n 3 3 Thought Question Given the properties of the above two studies, which study would have the LARGER confidence intervals? c(study1, study2) n=3 n=3 Study 1 Study 2 rep(c("study 1", "Study 2"), c(3, 3))

8 Thinking Through Suppose that we have two different datasets with the following properties: Thinking Through > study1 <- rnorm(3, 1, 1) > study2 <- rnorm(3, 1, 1) > plotmeans(c(study1, study2) ~ rep(c("study 1", + "Study 2"), c(3, 3))) Study 1 Study 2 mean 1 1 SD 1 1 n 3 3 Thought Question Given the properties of the above two studies, which study would have the LARGER confidence intervals? c(study1, study2) n=3 n=3 Study 1 Study 2 rep(c("study 1", "Study 2"), c(3, 3)) f() Specific Normal Distributions N(1,1) N(1,4) N(12,1)

9 The standard normal distribution The standard normal distribution is the one with µ = and σ = 1. We often use Z to denote it, i.e. Z N (, 1 2 ). Its density function is often written φ(z) and φ(z) = 1 2π e z2 /2 < z < Evaluating the Normal Distribution As given from the previous slide, we assume that the normal distribution has a structure such that mean meadian mode. We can also evaluate the shape of our distribution relative to it s deviation from the standard normal distribution through multiple means. - a measure of shape determining whether or not the left half looks like the right half - a measure of shape determining relative hieght to width Gaussian probability density plot measures typically range from -3 to +3 A skewness of means that the left half looks eactly like the right half Positively skewed means that the mean is influenced by outliers making it seem larger than the relative mean of the population from which the sample was drawn n ( i n skew = n (n 1)(n 2) SD ) 3 = n (n 1)(n 2) z 3 i

10 Eamples - Normal Distribution Eamples - Positively Skewed Distribution Histogram for Normally Distributed Data QQ plot for a Normal Distribution Histogram for Positively Skewed Data QQ plot for Positively Skewed Data rnorm(1, 1, 15) rskt(2, 2, 2) rnorm(1, 1, 15) qnorm rskt(2, 2, 2) qnorm Eamples - Negatively Skewed Distribution Histogram for Negatively Skewed Data rskt(2, 2, 2) rskt(2, 2, 2) QQ plot for Negatively Skewed Data qnorm

11 measures typically range from -3 to +3 A kurtosis of means that your distribution has the same relative height to width properties as a normal distribution Positive kurtosis means that your distribution is leptokurtic (taller and skinnier) Negative kurtosis means that your distribution is platykurtic (shorter and fatter) n(n + 1) n ( ) i 4 3(n 1)2 kurt = (n 1)(n 2)(n 3) SD (n 2)(n 3) f() Graphical Eamples of Although all of these theoretically could be normal, they illustrate leptokurtic and platykurtic distributions Normal Platykurtic Leptokurtic

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