Chapters 2 & 4

Aron, Coups, & AronStatistics for the Behavioral & Social Sciences- Chapter 2 and 4z Scores & the Normal Curve

The normal distribution and standard deviationsMean-1+1In a normal distribution:Approximately 68% of scores will fall within one standard deviation of the mean

The normal distribution and standard deviationsMean+1-1+2-2In a normal distribution:Approximately 95% of scores will fall within two standard deviations of the mean

The normal distribution and standard deviationsMean+2+1-1-2+3-3In a normal distribution:Approximately 99% of scores will fall within three standard deviations of the mean

Using standard deviation units to describe individual scoresHere is a distribution with a mean of 100 and and standard deviation of 10:1001101209080-1 sd1 sd2 sd-2 sdWhat score is one sd below the mean?90120What score is two sd above the mean?

Using standard deviation units to describe individual scoresHere is a distribution with a mean of 100 and and standard deviation of 10:1001101209080-1 sd1 sd2 sd-2 sd1How many standard deviations below the mean is a score of 90?How many standard deviations above the mean is a score of 120?2

Z scoresz scores are sometimes called standard scoresHere is the formula for a z score:A z score is a raw score expressed in standard deviation units.What is a z-score?

What does a z-score tell us?z-score describes the location of the raw score in terms of distance from the mean, measured in standard deviationsGives us information about the location of that score relative to the “average” deviation of all scoresA z-score is the number of standard deviations a score is above or below the mean of the scores in a distribution.A raw score is a regular score before it has been converted into a Z score.Raw scores on very different variables can be converted into Z scores and directly compared.

Z-score DistributionMean of zeroZero distance from the meanStandard deviation of 1The z-score has two parts:The numberThe signNegative z-scores aren’t badZ-score distribution always has same shape as raw score

Computational Formulaz = (X –M)/SDScore minus the mean divided by the standard deviation

Steps for Calculating a z-scoreJacob spoke to other children 8 times in an hour, the mean number of times children speak is 12, and the standard deviation is 4, (example from text).To change a raw score to a Z score:Step One: Determine the deviation score.Subtract the mean from the raw score.8 – 12 = -4Step Two: Determine the Z score. Divide the deviation score by the standard deviation.-4 / 4 = -1

Using z scores to compare two raw scores from different distributionsYou score 80/100 on a statistics test and your friend also scores 80/100 on their test in another section. Hey congratulations you friend says—we are both doing equally well in statistics. What do you need to know if the two scores are equivalent? the mean?What if the mean of both tests was 75?You also need to know the standard deviationWhat would you say about the two test scores if the SDin your class was 5 and the SDin your friend’s class is 10?

Calculating z scoresWhat is the z score for your test: raw score = 80; mean = 75, SD= 5?What is the z score of your friend’s test: raw score = 80; mean = 75, S = 10?Who do you think did better on their test? Why do you think this?

Why z-scores?Transforming scores in order to make comparisons, especially when using different scalesGives information about the relative standing of a score in relation to the characteristics of the sample or populationLocation relative to meanRelative frequency and percentile

Fun facts about z scores Any distribution of raw scores can be converted to a distribution of z scoresthe mean of a distribution has a z score of ____?zeropositive z scores represent raw scores that are __________ (above or below) the mean?abovenegative z scores represent raw scores that are __________ (above or below) the mean?below

Computing Raw Score from a z-scoreFigure the deviation score.Multiply the Z score by the standard deviation.Figure the raw score.Add the mean to the deviation score.Formula for changing a Z score to a raw score: X= (Z)(SD)+M

Comparing Different Variables Standardizes different scores Example in text:Statistics versus English test performanceCan plot different distributions on same graph increased height reflects larger N

How Are You Doing?How would you change a raw score to a Z score?If you had a group of scores where M = 15 and SD = 3, what would the raw score be if you had a Z score of 5?

Normal Distributionhistogram or frequency distribution that is a unimodal, symmetrical, and bell-shapedResearchers compare the distributions of their variables to see if they approximately follow the normal curve.

The Standard Normal CurveUse to determine the relative frequency of z-scores and raw scoresProportion of the area under the curve is the relative frequency of the z-scoreRarely have z-scores greater than 3 (.26% of scores above 3, 99.74% between +/- 3)

Why the Normal Curve Is Commonly Found in NatureA person’s ratings on a variable or performance on a task is influenced by a number of random factors at each point in time.These factors can make a person rate things like stress levels or mood as higher or lower than they actually are, or can make a person perform better or worse than they usually would.Most of these positive and negative influences on performance or ratings cancel each other out.Most scores will fall toward the middle, with few very low scores and few very high scores.This results in an approximately normal distribution (unimodal, symmetrical, and bell-shaped).

The Normal Curve Table and Z ScoresA normal curve table shows the percentages of scores associated with the normal curve.The first column of this table lists the Z scoreThe second column is labeled “% Mean to Z” and gives the percentage of scores between the mean and that Z score.The third column is labeled “% in Tail.”.

Using the Normal Curve Table to Figure a Percentage of Scores Above or Below a Raw ScoreIf you are beginning with a raw score, first change it to a Z Score.Z = (X – M) / SDDraw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage.Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages.Find the exact percentages using the normal curve table.Look up the Z score in the “Z” column of the table.Find the percentage in the “% Mean to Z” column or the “% in Tail” column.If the Z score is negative and you need to find the percentage of scores above this score, or if the Z score is positive and you need to find the percentage of scores below this score, you will need to add 50% to the percentage from the table.Check that your exact percentage is within the range of your rough estimate.

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 1Draw a picture of the normal curve and shade in the approximate area of your percentage using the 50%–34%–14% percentages.We want the top 5%.You would start shading slightly to the left of the 2 SD mark.Copyright © 2011 by Pearson Education, Inc. All rights reserved

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 2Make a rough estimate of the Z score where the shaded area stops.The Z Score has to be between +1 and +2.Copyright © 2011 by Pearson Education, Inc. All rights reserved

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 3Find the exact Z score using the normal curve table.We want the top 5% so we can use the “% in Tail” column of the normal curve table.The closest percentage to 5% is 5.05%, which goes with a Z score of 1.64.Copyright © 2011 by Pearson Education, Inc. All rights reserved

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 4Check that your Z score is within the range of your rough estimate.+1.64 is between +1 and +2.Copyright © 2011 by Pearson Education, Inc. All rights reserved

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 5If you want to find a raw score, change it from the Z score.X = (Z)(SD) + MX = (1.64)(16) + 100 = 126.24Copyright © 2011 by Pearson Education, Inc. All rights reserved

Chapters 2 & 4

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Chapters 2 & 4