Normal Distributions and Standard Scores PSYC 214 Normal Distributions and Standard Scores

Agenda Percentiles Distributions Standard (-ization of) Scores Appendix B Examples

Few Reminders Rounding Greater than or less than > or < Usually round to the hundredth’s place (two numbers after decimal point e.g. 1.234 = 1.23, 1.235 = 1.24) Below 5 round down 5 and above round up Greater than or less than > or < Greater than > (e.g. 6 > 5) Less than < (e.g. 1 < 4) Alligator eats the biggest number

Percentile Rank Percentage of scores in the distribution that are at or below a specific value; Pxx Describes the position of a score In relation to other scores What type of measurement is this? Example: Baby at 90th percentile for height & weight is tall & heavy: 90% of other babies are shorter/lighter and only 10% are longer/heavier p. 101-105

95th percentile Baby

Another View (7 months)

My Nieces 70th Height and 57th Weight (4 yrs) 55th Height and 32nd Weight (18mths)

2 years 3 years 5 years

9 years, 11 years, and 7 years

How to find Pxx A score’s percentile: “If you were 45th from the bottom of 60 people, you’d be at the _____ percentile.” Score position/total number of scores X 100 = percentile 45/60 X 100 = 75th percentile or P75 If you scored 50th from the bottom of a class of 160 people, at what percentile would your score be?

Practice 23th from the bottom of 375 scores? What’s the percentile rank for a score that’s: 23th from the bottom of 375 scores? 7th from the end out of 34 scores? 95th from the bottom of 180 scores? What’s the position (from the bottom) for a score with a percentile rank of: P25 among 775 scores? P90 among 425 scores?

Percentile Rank as a z-Score But remember: Pxx can reflect an ordinal measure of relative standing (position relative to the group of scores) As a rank it does not take into consideration means or standard deviations (no math!) Standardized scores are often used to present percentile ranks, also referred to as z-scores They describe how far a given score is from the other scores, in terms of the z distribution

Purpose of z-Scores Identify and describe location of every score in the distribution Standardize an entire distribution Takes different distributions and makes them equivalent and comparable

Two distributions of exam scores

Locations and Distributions Exact location is described by z-score Sign tells whether score is located above or below the mean Number tells distance between score and mean in standard deviation units

Relationship of z-scores and locations

Traits of the normal distribution Sections on the left side of the distribution have the same area as corresponding sections on the right Because z-scores define the sections, the proportions of area apply to any normal distribution Regardless of the mean Regardless of the standard deviation

Normal Distribution with z-scores

z-Scores for Comparisons All z-scores are comparable to each other Scores from different distributions can be converted to z-scores The z-scores (standardized scores) allow the comparison of scores from two different distributions along

Properties of z scores Mean ALWAYS = 0 Standard deviation ALWAYS = 1 Positive z score is ABOVE the mean Negative z score is BELOW the mean

Standard Scores Give a score’s distance above or below the mean in terms of standard deviations Positive z-scores are always above the mean Negative z-scores are always below the mean Negative & positive z-scores only indicate directionality on the x-axis, not necessarily a change in the value of the score

Standard Scores An index of a score’s relative standing Standard (converted) scores are referred to as z-scores x = raw score µ = population mean σ = population SD Backward formula: x = z (σ) + µ Use this one to solve for x p.105

Equation for z-score Numerator is a deviation score Denominator expresses deviation in standard deviation units

Determining raw score from z-score Numerator is a deviation score Denominator expresses deviation in standard deviation units

Order of Operation Regular formula Backward formula Raw score (x) Std. score (z) % or Proportion Raw (X) z % or Proportion May not go directly from X to % or % to X – always go through z

Standard Normal Distribution 50% of total area under the curve +1 σ is where the line turns from concave to convex Symmetrical Asymptotic μ = 0 σ = 1.0 p.110, Fig. 4.1

Distributions Normal distribution: all normal distributions are symmetrical and bell shaped. Thus, the mean, median & mode are all equal! “Ideal world” bell curve μ = 0 σ = 1.0 Completely symmetrical Asymptotic

Standard Normal Distribution Fig 4.1

IQ Standard Normal Distribution 68% 95% 95% 99% 99% Genius Gifted Above average Higher average Lower average Below average Borderline low Low >144 130-144 115-129 100-114 85-99 70-84 55-69 <55 0.13% 2.14% 13.59% 34.13% Adapted from http://en.wikipedia.org/wiki/IQ_test

Shape of a Distribution Researchers describe a distribution’s shape in words rather than drawing it Symmetrical distribution: each side is a mirror image of the other Skewed distribution: scores pile up on one side and taper off in a tail on the other Tail on the right (high scores) = positive skew Tail on the left (low scores) = negative skew

Skewed Distributions Mean, influenced by extreme scores, is found far toward the long tail (positive or negative) Median, in order to divide scores in half, is found toward the long tail, but not as far as the mean Mode is found near the short tail. If Mean – Median > 0, the distribution is positively skewed. If Mean – Median < 0, the distribution is negatively skewed

Distributions Positively skewed distribution: tendency for scores to cluster below the mean p.90

Distributions Negatively skewed distribution: tendency for scores to cluster above the mean p.90

z z Appendix B Z or Z μ or μ p.617 Used to find proportions under the normal curve We only use Columns 1, 3 and 5 Column 3 gives proportion from the z-score towards either tail Column 5 gives proportion between z-score and the mean z Z or z μ or μ Z

Standard Scores Standard scores = z scores (in pop) µ = mu = population mean σ = sigma = population standard deviation x = z (σ) + µ (Backward formula)

Finding a proportion from a raw score 1: Sketch the normal distribution 2: Shade the general region corresponding to the required proportion 3: Using the “forward” formula compute the corresponding z-score from the raw score (x). 4: Locate the proportion in the correct column of the table

What proportion of people have an IQ of 85 or less? What about x < 70? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For IQ scores µ = 100 & σ =15

Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B What proportion of people have an IQ of 130 or above? Of x > 120? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For IQ scores µ = 100 & σ =15

Finding a z-score from a proportion 1: Sketch the normal distribution 2: Shade the general region corresponding to the required proportion 3: Locate the proportion in the correct column of the table 4: Identify corresponding z-score in Col.1 5: Calculate raw score (x) with “backwards” formula

What is the lowest IQ score you can earn and still be in the top 5% of the population? Step 1: draw curve Step 2: Appendix B Which column do you use? Step 3: use formula X = (z)(σ) + µ For IQ scores µ = 100 & σ =15

What proportion of test takers score between 300 and 650 on the SAT? Step 1: draw curve Step 2: use formula z = x – μ σ Step 3: Appendix B Which column do you use? For SAT scores µ = 500 & σ =100

Applications of z-Scores Any observation from a normal distribution can be converted to a z-score If we have an actual, theoretical, or hypothesized normal distribution of many means we can determine the position of one particular mean relative to the other means in this distribution This logic is the basis of many hypothesis tests

Z-score Practice Lecture Handout