z scores lecture chapter 2 and 4

Aron, Coups, & Aron Statistics for the Behavioral & Social

Sciences- Chapter 2 and 4

z Scores & the Normal Curve

Normal distribution

The normal distribution and standard deviations

Approximately 68% of scores will fall within one standard deviation of the mean

In a normal distribution:-1 +1

Mean


Approximately 95% of scores will fall within two standard deviations of the mean

In a normal distribution:

Mean

+1-1 +2-2


Approximately 99% of scores will fall within three standard deviations of the mean

In a normal distribution:

Mean

+2+1-1-2 +3-3

The Normal Curve

Using standard deviation units to describe individual scores

Here is a distribution with a mean of 100 and and standard deviation of 10:

100 110 1209080-1 sd 1 sd 2 sd-2 sd

What score is one sd below the mean? 90

What score is two sd above the mean? 120

Using standard deviation units to describe individual scores

Here is a distribution with a mean of 100 and and standard deviation of 10:

100 110 1209080-1 sd 1 sd 2 sd-2 sd

How many standard deviations below the mean is a score of 90? 1

2How many standard deviations above the mean is a score of 120?

Z scores

What is a z-score? A z score is a raw score expressed in standard deviation units.

z scores are sometimes called standard scores

SD

MXz

Here is the formula for a z score:

z-score describes the location of the raw score in terms of distance from the mean, measured in standard deviations

Gives us information about the location of that score relative to the “average” deviation of all scores

A 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.

What does a z-score tell us?

Mean of zeroZero distance from the mean

Standard deviation of 1The z-score has two parts:

The numberThe sign

Negative z-scores aren’t badZ-score distribution always has same

shape as raw score

Z-score Distribution

z = (X –M)/SD

Score minus the mean divided by the standard deviation

Computational Formula

Jacob 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

Steps for Calculating a z-score

Using z scores to compare two raw scores from different distributions

You 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 deviation

What would you say about the two test scores if the SD in your class was 5 and the SD in your friend’s class is 10?

Calculating z scoresWhat is the z score for your test: raw score = 80; mean = 75, SD = 5?

SD

MXz

1

5

7580

z

What is the z score of your friend’s test: raw score = 80; mean = 75, S = 10?

5.10

7580

z

Who do you think did better on their test? Why do you think this?

SD

MXz

Transforming scores in order to make comparisons, especially when using different scales

Gives information about the relative standing of a score in relation to the characteristics of the sample or populationLocation relative to meanRelative frequency and percentile

Why z-scores?

Fun facts about z scores

• Any distribution of raw scores can be converted to a distribution of z scores

positive z scores represent raw scores that are __________ (above or below) the mean?

above

negative z scores represent raw scores that are __________ (above or below) the mean?

below

the mean of a distribution has a z score of ____?

zero

Figure 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

Computing Raw Score from a z-score

Standardizes different scores Example in text:

Statistics versus English test performanceCan plot different distributions on same graph increased height reflects larger N

Comparing Different Variables

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?

histogram 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.

Normal Distribution

Use to determine the relative frequency of z-scores and raw scores

Proportion of the area under the curve is the relative frequency of the z-score

Rarely have z-scores greater than 3 (.26% of scores above 3, 99.74% between +/- 3)

The Standard Normal Curve

Why the Normal Curve Is Commonly Found in Nature

A 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 Scores

A 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.”

.

Z % Mean to Z % in Tail

.09 3.59 46.41

.10 3.98 46.02

.11 4.38 45.62

Normal Curve Table A-1

Using the Normal Curve Table to Figure a Percentage of Scores Above or Below a Raw Score

1. If you are beginning with a raw score, first change it to a Z Score.Z = (X – M) / SD

2. Draw 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.

3. Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages.

4. 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.

5. Check that your exact percentage is within the range of your rough estimate.

Copyright © 2011 by Pearson Education, Inc. All rights reserved

Example of Using the Normal Curve Table to Figure Z Scores and Raw Scores: Step 1

Draw 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.



Make a rough estimate of the Z score where the shaded area stops.The Z Score has to be between

+1 and +2.



Find 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.



Check that your Z score is within the range of your rough estimate. +1.64 is between +1 and +2.



If you want to find a raw score, change it from the Z score.X = (Z)(SD) + MX = (1.64)(16) + 100 = 126.24

z scores lecture chapter 2 and 4

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