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

The variance and standard deviation are numbers that describe how far, on the average, scores are from their mean, mu. zBut we often want additional detail about how scores will fall around their mean. zWe may also wish to theorize about how scores should fall around their mean.

Describing and theorizing about how scores fall around their mean. zFrequency distributions zStem and leaf displays zBar graphs and histograms zTheoretical frequency distributions

Frequency distributions # of acdnts Absolute Frequency Cumulative Frequency Cumulative Relative Frequency Cumulative frequencies show number of scores at or below each point. Calculate by adding all scores below each point. Cumulative relative frequencies show the proportion of scores at or below each point. Calculate by dividing cumulative frequencies by N at each point.

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 frequency distributions zExpected frequency = Theoretical relative frequency X 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.

Using the theoretical frequency distribution known as the normal curve

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 is a figural representation of a theoretical frequency distribution. zThe frequency distribution represented by the 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.

NOTE: Since the curve is symmetrical around the mean, whatever happens on one side of the curve is exactly mirrored on the other side. This also means that the mean, the median and the mode are all the same in a normal distribution

The normal curve and Z scores zThe normal curve is the theoretical relative frequency 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

IMPORTANT CONCEPT: The proportion of the curve between any two points on the curve represents the theoretical relative frequency (TRF) of scores between those points.

Area of the curve between two points = proportion of scores between those points zFor example, if the area of the curve between two points is 46.32% of the curve, we would expect to find a proportion of.4632 of the scores between those two 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 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. Z Score Proportion mu to Z

Another important concept: Most scores are close to the mean! So if you have two equal sized intervals, the one closer to mu contains a higher proportion of scores What proportion of scores falls in the interval between Zs of -.50 to +.50 (an interval of one standard deviation right around the mean)? =.3830 (almost 40%) Note: this is the one-standard-deviation-wide interval with the highest proportion anywhere on the curve. Note that almost 40% of a population should score within half a standard deviation from the mean. Proportion = =.2586Proportion = =.2586

Intervals further from mu What proportion of scores falls in the interval between Zs of 0.00 to (an interval of one standard deviation starting at the mean, but not right around it)? This one can be read directly from the table (It is a little over a third) What proportion of scores falls in the interval between Zs of +.50 to (an interval of one standard deviation a little further from the mean)? =.2417 This time we are down to less than a quarter of the population.

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

Areas between two points on the curve

470 USING THE Z TABLE - Proportion of the scores between a specific Z 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 of the scores in a population between two Z scores that are identical in size, but have opposite signs. FrequencyFrequency score Standard deviations Proportion mu to Z for =.1179 Proportion between +Z and -Z = =

The critical values of the normal curve zCritical values of a distribution show which symmetrical interval around mu contains 95% and 99% of the curve. zIn the Z table, the critical values are starred and shown to three decimal places z95% (a proportion of.9500) is found between Z scores of –1.960 and z99% (a proportion of.9900) is found between Z scores of –2.576 and

-1.06 USING THE Z TABLE - Proportion of scores between a two different Z scores on opposite sides of the mean. (ADD THE TWO PROPORTIONS!). 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 of scores between two Z scores on the same side of the mean. (Subtract the smaller proportion from the larger one.) 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 %

Expected Frequencies

Obtaining expected frequencies (EF) from the normal curve. zBasic rule: To find an expected frequency, multiply the proportion of scores expected in the part of the curve by the total N. Expected frequency = theoretical relative frequency x N.

Expected frequencies are another least squared, unbiased prediction. Expected frequencies usually must be wrong, as they are routinely written to two decimal place. For example, it is impossible to actually find 65 hundreths of a score anywhere. So, expected frequencies are another set of least squared, unbiased predictions. Such predictions can be expected to be wrong, but close.

EF=TRF x N zIn the examples that I’ve solved that follow, let’s assume we have a population of size 300 (N=300) zTo find the expected frequency, compute the proportion of the curve between two specific Z scores, just as we have been doing. zThen multiply that proportion (also called the theoretical relative frequency or TRF) by N.

Expected frequency = theoretical relative frequency x number of participants (EF=TRF*N). TRF from mean to Z = -.30 = If N = 300: EF=.1179*300 = FrequencyFrequency Standard deviations Proportion mu to Z for =.1179 EF=.1179x300 = 35.37

Expected frequencies below a specific Z score

EF below a score. zThis is the opposite of expected frequencies above a score. It is like asking the EF between your Z score and the entire half the curve (50% or.5000) that lies below the mean. If Z is above mu, TRF is between two Z scores on opposite sides of the mean. To get TRF, add half of the curve (.5000) to the area from mu to Z. To get EF, then multiply TRF by N. zIf Z is below mu, TRF is between two Z scores on the same side of the mean. To get TRF, subtract the area from mu to Z from half of the curve (.5000). To get EF, then multiply TRF by Z

If N = 300, what is the EF of scores below a Z of zExpected frequency below a score: If Z is above mu, to get TRF, add half of the curve (.5000) to the area from mu to Z. TRF below Z = is =.8413.

If N = 300: EF=.8413 x 300 = FrequencyFrequency inches Proportion =.5000 up to mean Standard deviations for 1 SD =.8413

Percentile rank

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 below (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 to Z =.4821 yProportion at or below Z = =.0179 yPercentile =.0179 x 100 = 1.79 = 2 nd percentile

Percentile Rank is the percent of the population you score as well as or better = Theoretical Relative Frequency below your Z score times 100. What is the percentile rank of someone with a Z score of FrequencyFrequency inches Percentile:.5000 up to mean Standard deviations = x 100 =84.13 =84 th percentile

A rule about rounding 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 and below 1, 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

Below the 1 st percentile and above the 99 th : Don’t round! zWhat percentile are you at if your Z score is +3.04? zArea mu to Z = zSince Z is above the mean, add proportion mu to Z to.5000 zPercentile = ( )*100 = zAbove 99 th percentile, DON”T ROUND! zThe answer is the th percentile