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 them (SS), divide by N (  2 ) and take a square root(  ). Example: Scores on a Psychology quiz Student John Jennifer Arthur Patrick Marie X78357X78357  X = 30 N = 5  = 6.00 X -   (X-  ) = 0.00 (X -  )  (X-  ) 2 = SS =  2 = SS/N = 3.20  = = 1.79

Stem and Leaf Display zReading time data Reading Time Leaves 5,5,6,6,6,6,8,8,9 0,0,1,2,3,3,3 5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,9 0,0,1,2,3,3,3,3,4,4,4 5,5,5,5,6,6,6,8,9,9 0,0,0,1,2,3,3,3,4,4 5,6,6,6 0,1,1,1,2,3,3,4 6,6,8,8,8,8,8,9,9,9 0,1,1,1,2,2,2,4,4,4,4 i =.05 #i = 10

Transition to Histograms

Histogram of reading times Reading Time (seconds) FrequencyFrequency

Normal Curve

Principles of Theoretical Curves zExpected frequency = Theoretical relative frequency * N zExpected frequencies are your best estimates because they are closer, on the average, than any other estimate when we square the error. zLaw of Large Numbers - The more observations that we have, the closer the relative frequencies should come to the theoretical distribution.

The Normal Curve zDescribed mathematically by Gauss in So it is also called the “Gaussian”distribution. It looks something like a bell, so it is also called a “bell shaped” curve. zThe normal curve really represents a histogram whose rectangles have their corners shaved off with calculus. zThe normal curve is symmetrical. yThe mean (mu) falls exactly in the middle. y68.26% of scores fall within 1 standard deviation of the mean. y95.44% of scores fall within 2 standard deviations of the mean. y99.74% of scores fall within 3 standard deviations of mu.

The normal curve and Z scores zThe normal curve is a theoretical distribution that underlies most variables that are of interest to psychologists. zA Z score expresses the number of standard deviations that a score is above or below the mean in a normal distribution. zAny point on a normal curve can be referred to with a Z score

The Z table and the curve zThe Z table shows the normal curve in tabular form as a cumulative relative frequency distribution. zThat is, the Z table lists the proportion of a normal curve between the mean and points further and further from the mean. zThe Z table shows only the cumulative proportion in one half of the curve. The highest proportion possible on the Z table is therefore.5000

Why does the Z table show cumulative relative frequencies only for half the curve? zThe cumulative relative frequencies for half the curve are all one needs for all relevant calculations. zRemember, the curve is symmetrical. zSo the proportion of the curve between the mean and a specific Z score is the same whether the Z score is above the mean (and therefore positive) or below the mean (and therefore negative). zSeparately showing both sides of the curve in the Z table would therefore be redundant and (unnecessarily) make the table twice as long.

KEY CONCEPT The proportion of the curve between any two points on the curve represents the relative frequency of scores between those points.

With a little arithmetic, using the Z table, we can determine: The proportion of the curve above or below any Z score. Which equals the proportion of the scores we can expect to find above or below any Z score. The proportion of the curve between any two Z scores. Which equals the proportion of the scores we can expect to find between any two Z scores.

Normal Curve – Basic Geography FrequencyFrequency Measure The mean One standard deviation | | | | | | Z scores | | | Percentages Standard deviations

The z table The Z table contains a column of Z scores coordinated with a column of proportions. The proportion represents the area under the curve between the mean and any other point on the curve. The table represents half the curve Z Score Proportion mu to Z

Common Z scores – memorize these scores and proportions Z Proportion Score mu to Z (* 2 = 99% between Z= –2.576 and Z= ) ( * 2 = 95% between Z= –1.960 and Z= )

470 USING THE Z TABLE - Proportion between a score and the mean. FrequencyFrequency score Standard deviations Proportion mu to Z for =.1179 Proportion score to mean =.1179

470 USING THE Z TABLE - Proportion between score FrequencyFrequency score Standard deviations Proportion mu to Z for =.1179 Proportion between +Z and -Z = =

470 USING THE Z TABLE – Proportion of the curve above a score. FrequencyFrequency score Proportion above score Standard deviations Proportion mu to Z for.30 =.1179 Proportion above score = =.6179

-1.06 USING THE Z TABLE - Proportion between score and a different point on the other side of the mean. FrequencyFrequency Percent between two scores Z scores Proportion mu to Z for =.3554 Proportion mu to Z for.37 =.1443 Area Area Add/Sub Total Per Z 1 Z 2 mu to Z 1 mu to Z 2 Z 1 to Z 2 Area Cent Add %

+1.50 USING THE Z TABLE - Proportion between score and another point on the same side of the mean. FrequencyFrequency Percent between two scores Z scores Proportion mu to Z for 1.12 =.3686 Proportion mu to Z for 1.50 =.4332 Area Area Add/Sub Total Per Z 1 Z 2 mu to Z 1 mu to Z 2 Z 1 to Z 2 Area Cent Sub %

USING THE Z TABLE – Expected frequency = theoretical relative frequency * number of participants (EF=TRF*N). Expected frequency between mean and Z = If N = FrequencyFrequency Standard deviations Proportion mu to Z for =.1179 EF=.1179*300 = 35.37

USING THE Z TABLE – Expected frequency = theoretical relative frequency * number of participants (EF=TRF*N). Expected frequency above Z = -.30 if N = 300. EF=.6179 * 300 = Standard deviations -.30 FrequencyFrequency Proportion mu to Z for.30 =.1179 Proportion above score = =.6179

USING THE Z TABLE – Percentage below a score FrequencyFrequency inches What percent of the population scores at or under a Z score of Percentage = 50 % up to mean Standard deviations % for 1 SD = 84.13%

USING THE Z TABLE – Percentile Rank is the proportion of the population you score as well as or better than times 100. FrequencyFrequency inches What is the percentile rank of someone with a Z score of Percentile:.5000 up to mean Standard deviations = * 100 =84.13 =84 th percentile

Percentile rank is the proportion of the population you score as well as or better than times 100. The proportion you score as well as or better than is shown by the part of the curve to the left of your score.

Computing percentile rank yAbove the mean, add the proportion of the curve from mu to Z to yBelow the mean, subtract the proportion of the curve from mu to Z from yIn either case, then multiply by 100 and round to the nearest integer (if 1 st to 99 th ). yFor example, a Z score of –2.10 yProportion mu tg Z =.4821 yProportion at or below Z = =.0179 yPercentile =.0179 * 100 = 1.79 = 2 nd percentile

A rule about percentile rank zBetween the 1 st and 99 th percentiles, you round off to the nearest integer. zBelow the first percentile and above the 99 th, use as many decimal places as necessary to express percentile rank. zFor example, someone who scores at Z=+1.00 is at the 100( ) = = 84 th percentile. zAlternatively, someone who scores at Z=+3.00 is at the 100( )=99.87= th percentile. Above 99, don’t round to integers. zWe never say that someone is at the 0 th or 100 th percentile.

Calculate percentiles Z Area Add to.5000 (if Z > 0) Proportion Percentile Score mu to Z Sub from.5000 (if Z < 0) at or below st th th rd th