Inferential Statistics: Frequency Distributions & Z-Scores.

Inferential Statistics:

Frequency Distributions&

Z-Scores

Outline of Today’s Discussion1. 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

ResearchRepresentation

ResearchResults

ResearchConclusions

Abstraction

Data Analysis

MethodologyGeneralization

***

Frequency DistributionsZ-Scores / PercentilesEvaluating 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).

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss



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…


Which distribution has the largest standard deviation?


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

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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….


5 9 6 75 1 5 55 9 6 35 1 5 55 9 4 55 1 5 35 1 7 75 9 4 55 9 3 75 1 5 3

A B C D


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”)…


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


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)…


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)”


Positively Skewed Distribution&

The Measures of Central Tendency


Soon we will learn about a very important, positively skewed distribution called

The Chi-Square Distribution ().


Negatively Skewed Distribution&

The Measures of Central Tendency

Part 2

Z-scores

The Z-Score1. 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-Score1. 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-Score1. 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-Score1. 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 BetweenZ-Scores & Percentile Rank

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.


Question: What percent of the population falls between

z-scores of zero, and negative infinite?


Question: So, if your z-score is zero,

what’s your percentile rank?


Question: If your z-score = +1,



Question: If your z-score = +2,



Question: If your IQ is 85,



By convention, if a score is in the top 2.5 percentile, or the bottom 2.5 percentile,

Psychologists consider the score to besignificantly different from the mean.

Z-Scores & Percentile Rank1. 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 Rank1. 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 0.025 (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 Rank1. 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 0.84. These should produce familiar z-scores…


The 50th percentile = the ?? Z-score? The 84th percentile = the ?? Z-score?

Z-Scores & Percentile Rank1. 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…


Using the NORMSINV command in Excel,find the z-scores that correspond to

the 25th & 75th percentile.

Z-Scores & Percentile Rank1. 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 Rank1. 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 Rank1. 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?

AcknowledgmentsImages used in this educational presentation were obtained from Wikimedia Commons, in accordance with regulations regarding copyright, use, and dissemination.

http://commons.wikimedia.org/wiki/Main_Page

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Inferential Statistics: Frequency Distributions & Z-Scores.

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