Chapter 4 Translating to and from Z scores, the standard error of the mean and confidence intervals Welcome Back! NEXT.

Slides:



Advertisements
Similar presentations
Chapter 5 Some Key Ingredients for Inferential Statistics: The Normal Curve, Probability, and Population Versus Sample.
Advertisements

The Normal Curve and Z-scores Using the Normal Curve to Find Probabilities.
The Normal distributions BPS chapter 3 © 2006 W.H. Freeman and Company.
Chapter 6: Standard Scores and the Normal Curve
Chapter 9: The Normal Distribution
Chapter 3 The Normal Curve.
The standard error of the sample mean and confidence intervals
The standard error of the sample mean and confidence intervals
Chapter 3 The Normal Curve Where have we been? To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum.
Probability & Using Frequency Distributions Chapters 1 & 6 Homework: Ch 1: 9-12 Ch 6: 1, 2, 3, 8, 9, 14.
Chapter 4 Translating to and from Z scores, the standard error of the mean and confidence intervals Welcome Back! NEXT.
PSY 307 – Statistics for the Behavioral Sciences
Chapter 3 The Normal Curve Where have we been? To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum.
PPA 415 – Research Methods in Public Administration Lecture 5 – Normal Curve, Sampling, and Estimation.
Chapter 2 Frequency Distributions, Stem-and- leaf displays, and Histograms.
Chapter 3 The Normal Curve Where have we been? To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum.
The standard error of the sample mean and confidence intervals How far is the average sample mean from the population mean? In what interval around mu.
Chapter 4 Part 1 Translating to and from Z scores, the standard error of the mean, and confidence intervals around muT Welcome Back! NEXT.
Chapter 2 online slides Chapter 2 Frequency Distributions, Stem-and- leaf displays, and Histograms.
Chapter 2 Frequency Distributions, Stem-and-leaf displays, and Histograms.
Chapter 3 Z Scores & the Normal Distribution Part 1.
S519: Evaluation of Information Systems Social Statistics Chapter 7: Are your curves normal?
Chapter 1-6 Review Chapter 1 The mean, variance and minimizing error.
z-Scores What is a z-Score? How Are z-Scores Useful? Distributions of z-Scores Standard Normal Curve.
C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview Central Limit Theorem The Normal Distribution The Standardised Normal.
BPS - 5th Ed. Chapter 31 The Normal Distributions.
Chapter 4 Translating to and from Z scores, the standard error of the mean and confidence intervals Welcome Back! NEXT.
The standard error of the sample mean and confidence intervals How far is the average sample mean from the population mean? In what interval around mu.
Objectives (BPS 3) The Normal distributions Density curves
Measures of Central Tendency
Frequency Table Frequency tables are an efficient method of displaying data The number of cases for each observed score are listed Scores that have 0 cases.
Basic Statistics Standard Scores and the Normal Distribution.
Statistics Used In Special Education
The Normal Distribution The “Bell Curve” The “Normal Curve”
Probability & the Normal Distribution
16-1 Copyright  2010 McGraw-Hill Australia Pty Ltd PowerPoint slides to accompany Croucher, Introductory Mathematics and Statistics, 5e Chapter 16 The.
Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 1 PROBABILITIES FOR CONTINUOUS RANDOM VARIABLES THE NORMAL DISTRIBUTION CHAPTER 8_B.
Chapter 9 – 1 Chapter 6: The Normal Distribution Properties of the Normal Distribution Shapes of Normal Distributions Standard (Z) Scores The Standard.
Points in Distributions n Up to now describing distributions n Comparing scores from different distributions l Need to make equivalent comparisons l z.
 Frequency Distribution is a statistical technique to explore the underlying patterns of raw data.  Preparing frequency distribution tables, we can.
Slide 1 © 2002 McGraw-Hill Australia, PPTs t/a Introductory Mathematics & Statistics for Business 4e by John S. Croucher 1 n Learning Objectives –Identify.
Review Ways to “see” data –Simple frequency distribution –Group frequency distribution –Histogram –Stem-and-Leaf Display –Describing distributions –Box-Plot.
Descriptive Statistics: Presenting and Describing Data.
Thursday August 29, 2013 The Z Transformation. Today: Z-Scores First--Upper and lower real limits: Boundaries of intervals for scores that are represented.
Chapter 6 The Normal Distribution. 2 Chapter 6 The Normal Distribution Major Points Distributions and area Distributions and area The normal distribution.
The Normal distributions BPS chapter 3 © 2006 W.H. Freeman and Company.
Chapter 4 & 5 The Normal Curve & z Scores.
Hand out z tables Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2015.
NORMAL DISTRIBUTION Chapter 3. DENSITY CURVES Example: here is a histogram of vocabulary scores of 947 seventh graders. BPS - 5TH ED. CHAPTER 3 2 The.
The Normal Curve & Z Scores. Example: Comparing 2 Distributions Using SPSS Output Number of siblings of students taking Soc 3155 with RW: 1. What is the.
THE NORMAL DISTRIBUTION AND Z- SCORES Areas Under the Curve.
Chapter 2: Frequency Distributions. Frequency Distributions After collecting data, the first task for a researcher is to organize and simplify the data.
The Normal Distribution Lecture 20 Section Fri, Oct 7, 2005.
The Normal distribution and z-scores
Psych 230 Psychological Measurement and Statistics Pedro Wolf September 16, 2009.
The Normal Distribution and Norm-Referenced Testing Norm-referenced tests compare students with their age or grade peers. Scores on these tests are compared.
Chapter 2 Describing and Presenting a Distribution of Scores.
Describing a Score’s Position within a Distribution Lesson 5.
The Normal Distribution Lecture 20 Section Mon, Oct 9, 2006.
The Normal Distributions.  1. Always plot your data ◦ Usually a histogram or stemplot  2. Look for the overall pattern ◦ Shape, center, spread, deviations.
The Normal Distribution
Chapter 6 The Normal Curve.
Normal Distributions and Standard Scores
Finding Probabilities
Theoretical Normal Curve
Practice A research was interested in the relation between stress and humor. Below are data from 8 subjects who completed tests of these two traits.
Measuring location: percentiles
Quantitative Methods PSY302 Quiz Normal Curve Review February 6, 2017
Z Scores & the Normal Distribution
Practice #7.7 #7.8 #7.9. Practice #7.7 #7.8 #7.9.
Presentation transcript:

Chapter 4 Translating to and from Z scores, the standard error of the mean and confidence intervals Welcome Back! NEXT

Where have we been We have learned that you can describe how a distribution of numbers falls around its mean. That description can be summary numbers, such as the population mean (mu), the variance or standard deviation (sigma2 or sigma) and how many scores there are in the population (N).

Equivalence of figural and tabular displays When we want more detail about how scores fall around their mean, we can use tabular displays (such as frequency distributions) or figural displays (such as histograms) as a description. Both figures and tables can portray the tally, the way scores fall around their mean, in several ways. These included displaying absolute and relative frequencies in simple or cumulative forms. Both theoretical and actual distributions can be displayed as figures or tables

The most important theoretical relative frequency distribution is the normal curve, also called the Z curve. The normal curve (also called the Gaussian distribution or the normal error curve or the Z curve) is the most important theoretical frequency distribution. Theoretical relative frequency distributions tell us how scores can be expected to fall around their mean. As figures, theoretical relative frequency distributions show the proportion of scores that fall in a specific range around the mean.

Tabular forms of relative frequency distributions show culumative relative frequencies The normal curve is symmetrical around the mean. The Z table displays one side of the normal curve. It shows the cumulative proportion of scores between the mean and a specific score as that score moves further and further from the mean. Z scores express number of standard deviations above or below the mean.

So the Z table shows the cumulative relative frequency of scores between the mean and a specific number of standard deviations above or below the mean. Here is what it looks like, with the odd numbered columns showing distance from the mean expressed as a Z score and even numbered columns showing the cumulative relative frequency of scores between mu and Z.

The z table Z Score Proportion mu to Z 0.00 0.01 0.02 0.03 0.04 . 1.960 2.576 3.90 4.00 4.50 5.00 Proportion mu to Z .0000 .0040 .0080 .0120 .0160 . .4750 .4950 .49995 .49997 .499997 .4999997 The Z table contains pairs of columns: columns of Z scores coordinated with columns of proportions from mu to Z. The columns of proportions show the proportion of the scores that can be expected to lie between the mean and any other point on the curve. The Z table shows the cumulative relative frequencies for half the curve.

The Z table and normal curve are just different ways of presenting the same theoretical frequency distribution. The normal curve is like a histogram whose intervals have been made very, very tiny. In fact, it is a frequency polygon, in which lines connect the midpoints of each interval. You can make the intervals infinitely small, If you do that and connect the midpoints of those intervals, the result is a smooth line, the normal curve. If you have studied integral calculus you know precisely how this works. If not, to keep it parallel to the Z table, you can think of it as a smoothed out histogram with one hundred divisions for each standard deviation.

Normal Curve – Basic Geography F r e q u n c y The mean One standard deviation 3 2 1 0 1 2 3 Standard deviations Measure -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 Z scores Percentages |---34.13--|--34.13---| |--------47.72----------|----------47.72--------| |--------------49.87-----------------|------------------49.87------------|

Where we have been: Z scores If you know the proportion from the mean to the score, then you can easily calculate: The proportion above or below the score. The percentile rank equivalent. The proportion of scores between two Z scores. The expected frequency of scores between two Z scores

Z scores are the scores! Z scores represent standard deviations above and below the mean. Positive Z scores are scores higher than the mean. Negative Z scores are scores lower than the mean. If you know the mean and standard deviation of a population, then you can always convert a raw score to a Z score. If you know a Z score, the Z table will show you the proportion of the population between the mean and that Z score.

Symbols to memorize Mu Sigma Standard error of the mean Mean of a sample

Z scores are the scores! Raw scores to Z scores If we know mu and sigma, any score can be translated into a Z score: Z = score - mean standard deviation = X -  

Z scores to other scores Conversely, as long as you know mu and sigma, a Z score can be translated into any other type of score: Score =  + ( Z *  )

Calculating z scores Z = score - mean standard deviation What is the Z score for someone 6’ tall, if the mean is 5’8” and the standard deviation is 3 inches? Z = 6’ - 5’8” 3” = 72 - 68 3 = 4 3 = 1.33

If you know a Z score, you can determine theoretical relative frequencies and expected frequencies using the Z table. You often start with raw or scale scores and have to convert them to Z scores. Scale scores are public relations versions of Z scores, with preset means and standard deviations.

Production F r e q u n c y units Z score = ( 2100 - 2180) / 50 2100 = -80 / 50 = -1.60 3 2 1 0 1 2 3 Standard deviations units 2030 2330 2080 2280 2130 2180 2230 What is the Z score for a daily production of 2100, given a mean of 2180 units and a standard deviation of 50 units?

Concepts behind Scale Scores Scale scores are raw scores expressed in a standardized way. The most basic scale score is the Z score itself, with mu = 0.00 and sigma = 1.00. Raw scores can be converted to Z scores, which in turn can be converted to other scale scores. And Scale scores can be converted to Z scores, those Z scores, in turn, can be converted to raw scores.

You need to memorize these scale scores Z scores have been standardized so that they always have a mean of 0.00 and a standard deviation of 1.00. Other scales use other means and standard deviations. Examples: IQ -  =100;  = 15 SAT/GRE -  =500;  = 100 Normal scores -  =50;  = 10

To find percentiles or expected frequencies with scale scores, translate to Z scores and then use the Z table

For example: To solve the problem below, convert an SAT Score to a Z score, then use the Z table as usual. Proportion mu to Z for Z score of -.30 = .1179 F r e q u n c y Z score = ( 470 - 500) / 100 = -30 / 100 470 Proportion below score = .5000 - .1179 = . 3821 = 38.21% = -0.30 3 2 1 0 1 2 3 Standard deviations score 200 800 300 700 400 500 600 What percentage of test takers obtain a verbal score of 470 or less, given a mean of 500 and a standard deviation of 100?

SAT to percentile – first transform to a Z scores If a person scores 592 on the SATs, what percentile is she at? SAT  (X-)  (X-)/  592 500 92 100 0.92 Proportion mu to Z = .3212 Percentile = (.5000 + .3212) * 100 = 82.12 = 82nd

Convert to IQ scores to Z scores to find the proportion of scores between two IQ scores. IQ scores have mu = 100 and sigma = 15. What proportion of the scores falls between 85 and 115? Z score = (85 - 100) / 15 = -15 / 15 = -1.00 Z score = (115 - 100) / 15 = 15 / 15 = 1.00 Proportion = .3413 + .3413 = .6826 What proportion of the scores falls between 95 and 110? Z score = (95 - 100) / 15 = -5 / 15 = -0.33 Z score = (110 - 100) / 15 = 10 / 15 = 0.67 Proportion = .2486 + .1293 = .3779

NOTICE: Equal sized intervals, close to and further from the mean: More scores close to the mean! Given mu = 100 and sigma = 15, what proportion of the population falls between 95 and 105? Z score = (95 - 100) / 15 = -5 / 15 = -.33 Z score = (115 - 100) / 15 = 5 / 15 = .33 Proportion = .1293 + .1293 = .2586 What proportion of the population falls between 105 and 115? Z score = (105 - 100) / 15 = 5 / 15 = 0.33 Z score = (115 - 100) / 15 = 105/ 15 = 1.00 Proportion = ..3413 - .1293 = .2120

PROPORTIONS BETWEEN TWO SCALE SCORES Answer the following question without the table: Which of the following 10 point ranges will have the higher proportion of scores? 1. IQ scores of 115 to 125 2. IQ scores of 130 to 140

Without the table, the interval closer to the mean will have more of the scores.

Answer the following question with the table: What proportion of the scores fall into each of the 10 point ranges? 1. IQ scores of 115 to 125 2. IQ scores of 130 to 140

Proportions IQ scores of 115 and 125 = Z scores of 115-100/15 = 1.00 125-100/15=1.67 Proportion Z1-Z2 = .4525 -.3413=.1112 IQ scores of 130 and 140 = Z scores of 130-100/15 = 2.00 140-100/15=2.67 Proportion Z1-Z2 = .4962 -.4772=.0190

Convert IQ scores of 120 & 80 to percentiles. Now let’s compute percentile rank equivalents of IQ scores: First translate to Z scores Convert IQ scores of 120 & 80 to percentiles. X  (X-)  (X-)/  120 100 20.0 15 1.33 80 100 -20.0 15 -1.33 Now do the translation from Z score to percentile rank: mu-Z = .4082, .5000 + .4082 = .9082 = 91st percentile, Similarly 80 = .5000 - .4082 = 9th percentile Convert an IQ score of 100 to a percentile. An IQ of 100 is right at the mean and that’s the 50th percentile.

What is the percentile equivalent of a GRE score of 375? Note: GRE is the same as SAT. mu = 500, sigma = 100

GRE score to percentile rank Z=375-500/100 = -1.25 Proportion between mean and Z = .3944 We are below the mean, so to find percentile rank subtract .3944 from .5000, multiply by 100 and round if between the 1st & 99th percentiles. .5000-.3944 = .1056 .1056*100 = 10.56 round to 11th percentile

Going the other way – Z scores to scale scores Remember: Score =  + ( Z *  )

Convert Z scores to IQ scores: Individual scale scores get rounded to nearest integer. Z  (Z*)  IQ= + (Z * ) +2.67 15 40.05 100 +2.67 15 +2.67 15 40.05 100 140 +2.67 +2.67 15 40.05 -.060 15 -9.00 100 91

Tougher problems – like online quiz or midterm

If someone scores at the 58th percentile on the verbal part of the SAT, what is your best estimate of her SAT score?

Percentile to Z score to scale score If someone scores at the 58th percentile on the SAT-verbal, what SAT-verbal score did he receive? 58th Percentile is above the mean. This will be a positive Z score. The mean is the 50th percentile. So the 58th percentile is 8% or a proportion of .0800 above mu. So we have to find the Z score that gives us a proportion of .0800 of the scores between mu and Z. Look at Column 2 of the Z table on page 54. Closest Z score for area of .0800 is 0.20 Z  (Z*)  SAT= + (Z * ) 0.20 100 20 500 520

Slightly tougher –below the mean

Percentile to Z score to scale score If someone scores at the 38th percentile on the SAT-verbal, what SAT-verbal score did he receive? 38th percentile is below the mean. This will be a negative Z score. The mean is the 50th percentile. So the 38th percentile is 12% or a proportion of .1200 below mu. So we have to find the Z score that gives us a proportion of .1200 of the scores between mu and Z. Look at Column 2 of the Z table on page 54. Closest Z score for area of .1200 is 0.31. Z is negative Z  (Z*)  SAT= + (Z * ) -0.31 100 -31 500 469

You solve one, if someone scores at the 20th percentile, what is their IQ score

Always use Z scores to translate scores To go from a raw score to a scale score, you do the translation by turning the raw score into a Z score. Then you translate the Z score into a scale score,. This is one part of a general rule, you can transform any kind of score to any other kind of score when you know mu and sigma of both scores and therefore can use Z scores as your translator.

Do this double translation. On the GRE-Advanced Psychology exam, there are 225 questions. The mean is 125.00 correct with a standard deviation of 12.00. John gets 116 correct. What is his GRE score on this test?

On the GRE-Advanced Psychology exam, there are 225 questions. The mean is 125.00 correct with a standard deviation of 12.00. John gets 116. What is his GRE score on this test? Raw  (X- )  Scale Scale Scale score (raw) (raw) (raw) Z   score 116 125.00 -9.00 12.00 -0.75 500 100 425