Chapter 6 Notes
Chapter 6 Notes PSYCH 2220 - 0020
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This 14 page Class Notes was uploaded by Emma Dahlin on Thursday October 1, 2015. The Class Notes belongs to PSYCH 2220 - 0020 at Ohio State University taught by Anna Yocom in Summer 2015. Since its upload, it has received 51 views. For similar materials see Data Analysis in Psychology in Psychlogy at Ohio State University.
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Date Created: 10/01/15
The Bell Curve is Born 1769 I gt ill a Iquot 1 g A Modern Normal Curve Development of a Normal Curve Sample of 5 30 Frequency 1 5 1 7L Hi 27quot I 5000 5500 6000 6500 7000 7500 8000 Development of a Normal Curve Sample of 30 Frequency quotL 7 00 0 5500 6000 6500 7000 7500 8000 Height Development of a Normal Curve Sample of 140 4D an Friaquenqir 23 1393 5009 5500 b EELDID FELDEI FEJJEI 3000 0 As sample size increases the shape of the distribution becomes more like normal curve 0 Can you think of variables that might be normally distributed 0 IQ scores weight height reaction times 0 Eye color would NOT work bc its nominal 0 Nominal categorical variables cannot be normally distributed Standardization 2 Scores and the Normal Curve Standardization allows comparisons 0 Comparing zscore5 of standard deviations a score is from mean 0 2 distribution normal distribution of standardized scores Who Cares Most DVs are assumed to be normally distributed 0 If a variable is normally distributed we can know stuff about likelihood of occurrence 0 Most of our stats are based on assumption of normality The 2 Distribution 0 A z score can tell you 2quot 1 3 CI 3 1 1 standard Mean 1 standard deviation deviation below the above the mean mean Linear Transformations Small set of data 0 Mean607 0 Std dev162 If you add constant to each score X4 the mean would increase but the standard deviation would stay the same 0 If you subtract a constant from each score X6 the mean would decrease but the standard deviation would stay the same More Transformations If you divide scores by a constant what happens to mean and standard deviation 0 Making scores closer together 0 Mean would decrease and standard deviation would decrease split in half Transforming Raw Scores to 2 Scores 0 In order to use the Tables 81 we need mean0 std dev1 o p O o 1 0 Step 1 Subtract the mean of the population from the raw score 0 Step 2 Divide by the standard deviation of the population X ll 0 zscore of std dev above or below the mean Z iii 3 MW Transformations Mean GE 5 E 5 4 Stddev 39ll 9 5 T 5 5 5 4 939 5 5 Convert raw aceres into their zaeonres erc fes Mean StdDev 1 ee 1 23 131 nee D4 Transforming 2 Scores into Raw Scores 0 Step 1 Multiply the z score by the standard deviation of the population 0 Step 2 Add the mean of the population to this product XZOu in More Transformations MEEHl39FE 5 F B 6 4 Stddev 1152 5 7 E 5 Gdnvert ESSEGTEE into raw 5 4 9 5 5 scores gamma 53 119 04 l2 MEWEU 11 ee 53 anal 56 51211131 1 EE 423 131 ea 04 Example x1811626079 Comparing Apples and Oranges o If we can standardize raw scores on two different scales converting both scores to z scores we can then compare the scores directy EXAMPLE 0 Two different labs populations are measuring reaction times 0 Lab A measured reaction time after 1 drink of alcohol population mean was 21 seconds with a std deviation of 03 I Larry s reaction time was 14 seconds 0 Lab B measured reaction time after 2 drinks of alcohol The population mean was 31 seconds with a std deviation of 06 Bill s reaction time was 15 seconds 0 When using a standardized scale who has the larger slower reaction time o Larry39z1421O3 233 0 Bill z1531O6 267 0 Larry has the larger slower reaction time 233gt267 Transforming 2 Scores into Percentiles 0 Z scores tell you Where a value ts into a normal distribution 0 Based on the normal distribution there are rules about where scores with a 2 value will fall and how it will relate to a percentile rank 0 You can never use the area under the normal curve to calculate percentiles for any score The Normal Curve and Approximate Percentages 34 3 2 14 1W 2 3 2 1 0 1 2 3 o What percentage falls within 1 std deviation of mean 68 o What percentage falls within 2 std deviation of mean 96 o What percentage is less than the mean 50 o If you have a raw score equal to z 1 what is your percentile score 16th 0 What about 2 1 84th 0 z 15 91 o z 025 585 0 z 25 1 2 distribution always has a mean of O and a standard deviation of 1 The Normal Curve and Approximate Percentages 3 4 00 4 We 2 9X 1 4 39 1 4 00 WE If the population has a mean of 125 and std deviation of 10 if you are at the 16th percentile what would your score be 115 z 1 o If your score is 135 what is your percentile z1 84th percentile everything below If the population mean score on a quiz is 10 and the standard deviation is 2 o If a student s score is 8 what is z 1 If a student s z score is 14 what is the raw score 128 If a student s scores at the 84th percentile what is her raw score 12 zscore 1 Would you expect someone to have a score of 20 No What percentage of scores fall between O5 and 15 58 The Central Theorem Distribution of sample means is normally distributed even when population from which it was drawn is not normal A distribution of means is less variable than a distribution of individual scores Creating a Distribution of Scores These distributions were obtained by drawing from the same population 30 individual scores Frequency 5 l J i i Ml l l 393 J r l 59 53 54 55 5t 5 5399 ECI E11 IlaE 53 I314 5 Hi Elquot ENE Estzquot 33 3 1 T2 T3 T4 5395 El 3quot Height in in I135 30 mean scores Pregnancy 51 54quot fall 5quot 58 5quot til El EIE 4323 iiiquot b5 s El iiiE lbquot EF39EI TI 5393 T3 T4 3395 F39Ei 3quotquot Height in in has 0 There are less outliers in second graph bc there is less variability principle of averages it will all go towards middle Distribution of Means 0 Mean of the distribution tends to be mean of the population 0 Mean 1 0 Standard deviation of a distribution tends to be less than the standard deviation of the population of scores 0 Standard error standard deviation of the distribution of 039 means W o Smaller than std deviation takes on smaller value as N increases 0 Smaller variabilitylarger N OM Using the Appropriate Measure of Spread Distribution let means Free ueihegir Distribution of 39 Sheree 52 53 54 55 5f 5 58 55 EB 51 52 E33 54 55 Eat En ES 55 TD T1 Iquot TF3 T4 T5 H Ti Height in inthtee TABLE lPt tFt39 M ETEHS FDR DBE39IJ39HILJ TI39IGNE It39Z EIF SCGHEE UElFtBUE MEANS Wheh w determine the eeremetere ei e tietrihutiem we muet eeheitier whether tih dietrihutihh te E39 ijE t meehe er eeeree EtttllEtZtL EttMEDL DilETFilEtlTltitt FtiFi MEAN FFt SPREAD NAME FDR SPHEED Eteree 1e er Standard deeietieh theehe in my Eteheertl errer 0 Distribution of means follows central limit theorem 0 Shape of distribution approximates normal curve IF 0 Pop of scores has normal shape 0 Size of each sample of distribution is n30 A severely skewed distributien elf scares in a pepulatien a A less seue re 3 skewed distribution of means using samples of 2 from the same population bi Fl nermal distributien elf means using samples sf 5 frem the same population it The Mathematical Magic of Large Samples Example using Population Data A pizza delivery chain claims that it delivers its pizzas to any location within a 15 mile radius within an average of 30 minutes with a standard deviation of 12 minutes Supposed a researcher employed by a competitor orders 16 pizzas to be delivered to different locations with a 15 mile radius and computes a sample mean delivery time of 38 minutes What is this mean delivery time expressed as a 2 statistic 0 Standard error1243 o 2 38 303 267 0 what if sample size was 36 4 0 Normal Curve speci c bellshaped curve that is unimodal symmetric and de ned mathematically Sample sizeimportant in relation to normal curve Z As sample size increases approaches size of population the distribution more and more closely resembles a normal curve When data are normally distributed we can compare a score to an entire distribution of scores 0 To do this we convert raw score to standardized score Standardization way to convert individual scores from different normal distributions to a shared normal distribution with a known mean standard deviation and percentiles z score number of standard deviations a particular score is from the mean 0 Give us ability to convert any variable to a standard distribution allowing us to make comparisons among variables 0 2 distribution always has mean of O and standard deviation of 1 X M 039 To get raw score Xzap As long as we know mean and standard deviation of population we can do two things 1 calculate the raw score from its 2 score and 2 calculate the z score from its raw score Normal curve also allows us to convert scores to percentiles bc 100 of the population is represented under the bellshaped curve 0 Middle point50th percentile 2 distribution a normal distribution of standardized scores Standard normal distributionnormal distribution of z scores Standardized 2 distribution allows us to o Transform raw scores into standardized scores called 2 scores 0 Transform z scores back into raw scores 0 Compare 2 scores to each other even when the underlying raw scores are measured on different scales 0 Transform z scores into percentiles that are more easily understood Transforming 2 Scores into Percentiles z scores are useful bc 0 Give us a sense of where a score falls in relation to the mean of its population 0 Allow us to compare scores from different distributions 34 3 2 14 1W 2 3 2 l 0 1 2 3 Scores tell us how far a score is from the population mean in terms of the population standard deviation 0 2 distribution forms normal curve with unimodal symmetric shape The Central Limit Theorem Refers to how a distribution of sample means is a more normal distribution than a distribution of scores even when the population distribution is not normal 0 Repeated sampling approximates a normal curve even When the original population is not normally distributed 0 A distribution of means is less variable than a distribution of individual scores 0 Distribution of meanslja distribution composed of many means that are calculated from all possible samples of a given size all taken from the same population 0 not individual scores but rather means ofsampes of individual scores 0 When conducting hypothesis testing distribution of means is more useful than distribution of socres Creating Distribution of Means A distribution of means is more tightly Clustered has smaller standard deviation than a distribution of scores 0 Spread decreases and outliers are eliminated Characteristics of Distribution of Means OM Distribution of means has its own standard deviation which is smaller M mean of distribution of means O39M standard deviation of distribution of means 0 Typical amount that a sample mean varies from population mean Standard error the name for the standard distribution of distribution of means BC distribution of means is narrower than the distribution of scores it has a smaller standard deviation standard error 039 W 3 important characteristics 0 As sample size increases the mean of a distribution of means remains the same 0 The standard deviation standard error is smalerthan the standard deviation of distribution of scores As sample size increases the standard error becomes even smaller 0 The shape of the distribution of means approximates the normal curve if the distribution of the population of individual scores has a normal shape or if the size of each sample that makes up the distribution is at least 30 the central limit theorem Using Central Limit Theorem to Make Comparisons with 2 Scores 2 M pm om Because 2 score now represents a mean not an actual score it is often referred to as a 2 statistic Z statistic tells us how many standard errors a sample mean is from the population mean
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