Inferential Statistics: Frequency Distributions & Z-Scores

Outline of Today’s Discussion 1.Frequency Distributions: Overview 2.The Z-Score 3.Z-scores & Percentile Rank Please refrain from typing, surfing or printing during our conversation!

Part 1 Frequency Distributions: An Overview

The Research Cycle Real World Research Representation Research Results Research Conclusions Abstraction Data Analysis MethodologyGeneralization *** Frequency Distributions Z-Scores / Percentiles Evaluating Hypotheses

Frequency Distributions: Overview 1.Many naturally occurring phenomena are well described by a bell-shaped curve. 2.The bell-shaped curve is also called the Normal Distribution, or the Gaussian Distribution (after its founder, Gauss).

Frequency Distributions: Overview

1.The width of the normal distribution is an indication of the standard deviation. 2.“Fat” distributions have big standard deviations. 3.“Thin” distributions have small standard deviations…

Frequency Distributions: Overview Which distribution has the largest standard deviation?

Frequency Distributions: Overview Which distribution would have the smallest error bars if we were to plot the means and error bars on a bar-chart?

Gaussian “Width” & Error Bars Plus 1 S.D. Minus 1 S.D. Independent Variable Dependent Variable Ordinate starts at zero

Frequency Distributions: Overview 1.Now let’s further develop some intuitions about variability (variances, and standard deviations). 2.Each of the following four lists of numbers has a mean equaling 5. 3.Just by looking at the lists, indicate which list has the greatest standard deviation, and which one has the smallest standard deviation….

Frequency Distributions: Overview ABCD

1.Normal distributions are important in science because they allow for standardizing data across experiments and disciplines. 2.IQ scores, SAT scores, and height are just a few of the variables that are well described by the bell curve (i.e., “normally distributed”)…

Frequency Distributions: Overview We’ll return to z-scores later, this is just a peak to “prime” you.

Frequency Distributions: Overview 1.Not all distributions are symmetric. 2.Distributions that are not symmetric are said to be skewed. 3.Positively Skewed Distributions have “tails” to the right (the positive side of the axis). 4.Negatively Skewed Distributions have “tails” to the left (the negative side of the axis)…

Frequency Distributions: Overview Thinking about majoring in geoscience? Did you know that the average geoscience major who graduated from U.N.C in 1984ish earns $600,000 annually? What’s wrong with this picture? “The Thinker(s)”

Frequency Distributions: Overview Positively Skewed Distribution & The Measures of Central Tendency

Frequency Distributions: Overview Soon we will learn about a very important, positively skewed distribution called The Chi-Square Distribution (   ).

Frequency Distributions: Overview Negatively Skewed Distribution & The Measures of Central Tendency

Part 2 Z-scores

The Z-Score 1.If we have a normally distributed (Gaussian distributed) variable, and if there is not too much skew, we can describe a particular datum by it’s “z-score”. 2.A Z-score is a standard deviation from the mean. 3.Would someone tell us why z-scores are important? 4.Let’s see how we would compute a z-score by hand…

The Z-Score 1.Again, a Z-score is a standard deviation from the mean. 2.Let’s use IQ as an example. 3.If the standard deviation of IQ’s is 15 points, one z-score = 15 points above the mean; two z-scores = (2*15=) 30 points above the mean; three z-scores = (3*15=) 45 points above the mean; minus two z-scores = (-2*15=) -30 points below mean.

The Z-Score 1.There is a formula for computing z-scores, given a score (say, an IQ score), the mean, and the standard deviation, and vice versa. 2.Z = (raw score - mean) / standard deviation 3.Raw Score = mean + (Z * standard deviation) 4.Let’s do some examples…

The Z-Score 1.If the mean IQ = 100, with a standard deviation of 15, and Larry’s IQ score is 130, What’s Larry’s IQ in z-scores? 2.If the z-score associated with Craig’s IQ is -3, what’s his IQ score? 3.Note: if the standard deviation = 0, then the z- score is “undefined”. ( Can’t divide by zero! )

Part 3 The Relation Between Z-Scores & Percentile Rank

For ANY Gaussian distribution, 34% of the population falls between 0 and +1 z-score, and another 34% falls between 0 and -1 z-score.

Z-Scores & Percentile Rank Question: What percent of the population falls between z-scores of zero, and negative infinite?

Z-Scores & Percentile Rank Question: So, if your z-score is zero, what’s your percentile rank?

Z-Scores & Percentile Rank Question: If your z-score = +1, what’s your percentile rank?

Z-Scores & Percentile Rank Question: If your z-score = +2, what’s your percentile rank?

Z-Scores & Percentile Rank Question: If your IQ is 85, what’s your percentile rank?

Z-Scores & Percentile Rank By convention, if a score is in the top 2.5 percentile, or the bottom 2.5 percentile, Psychologists consider the score to be significantly different from the mean.

Z-Scores & Percentile Rank 1.Suppose we want to know what z-score is associated with, say, the bottom 2.5 percent of the population. 2.By “eye-balling it”, we know that the z-score should be approximately -2ish. 3.Let’s learn some commands in EXCEL to convert a given percentile rank (or a probability) into an exact z-score…

Z-Scores & Percentile Rank 1.In excel create one cell called “Proportion” (since a percentile rank is essentially a proportion). 2.We’re interested in the 2.5th percentile, which would be a proportion equaling (right?). 3.Label the neighboring cell “Z-Score”. 4.Beneath the label use the “=NORMSINV( )” function [or f(x)]. 5.NORMSINV is an abbreviation for Normal Standard Inverse. (Standard Gaussian ---> Mean = 0, SD = 1) 6.You can enter the desired proportion directly in the parentheses, or you can put a cell address there…

Z-Scores & Percentile Rank 1.Remember, this function is just like any other function…it’s a rule for turning one number into another number. 2.“You stick in a proportion (corresponding to percentile rank) and get out the corresponding z- score.” 3.Just to convince yourself, stick in some known values, like 0.5, and These should produce familiar z- scores…

Frequency Distributions: Overview The 50th percentile = the ?? Z-score? The 84th percentile = the ?? Z-score?

Z-Scores & Percentile Rank 1.Sometimes, psychologists want to know the inter- quartile range - the raw scores associated with the 25th and 75th percentiles. 2.Can you think of an example of when the inter- quartile range is used? 3.Here’s the inter-quartile range graphically…

Z-Scores & Percentile Rank Using the NORMSINV command in Excel, find the z-scores that correspond to the 25th & 75th percentile.

Z-Scores & Percentile Rank 1.Finally, let’s do it backwards!!! 2.That is, let’s say that we already have a z-score, but we want to know the corresponding percentile rank (i.e., the corresponding proportion)…

Z-Scores & Percentile Rank 1.We’ll use this command: “=NORMSDIST( )” 2.NORMSDIST is an abbreviation for Normal Standard Distribution. (Mean = 0, SD = 1) 3.The function “takes in a z-score” (in the parentheses), and then “spits out the corresponding proportion”. 4.Let’s do some examples…

Z-Scores & Percentile Rank 1.What percentile rank (i.e., proportion) is associated with a z-score of -2? 2.Here’s a tough one…What percentile rank is associated with an IQ of 145?

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