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PSYC 214 NORMAL DISTRIBUTIONS AND STANDARD SCORES Agenda Percentiles Distributions Standard (-ization of) Scores Appendix B Examples Few Reminders Rounding ◦ 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 th 95 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 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: o o o o 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 z X Numerator is a deviation score Denominator expresses deviation in standard deviation units Determining raw score from z-score z X so X z Numerator is a deviation score Denominator expresses deviation in standard deviation units Order of Operation Backward formula Regular formula Raw score (x) % or Proportion Std. score (z) Std. score (z) % or Proportion Raw score (x) 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) Symmetrical μ=0 2) Asymptotic σ = 1.0 +1 σ is where the line turns from concave to convex 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 95% 68% 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% 34.13% 13.59% 2.14% 0.13% 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 p.617 Appendix B 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 Z or z Column 5 gives proportion between z-score and the mean 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 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