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# Notes for 2/2/15-2/6/15 PSYC 3301

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This 10 page Class Notes was uploaded by Rachel Marte on Tuesday February 17, 2015. The Class Notes belongs to PSYC 3301 at University of Houston taught by Dr. Perks in Fall. Since its upload, it has received 84 views. For similar materials see Introduction to Psychological Statistics in Psychlogy at University of Houston.

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Date Created: 02/17/15

2215 Manipulating Formulas Degrees of Freedom Reporting Results and the Normal Curve Manipulating Formulas It is important to not only know useful statistical formulas but to be able to manipulate them in different ways based on What information you are given You may be asked to calculate the mean of a data set but you may also be asked to determine the sample size given the mean and the sum of the scores You should be able to identify Which formula to use based on What information is given to you and What the question is asking for Important Formulas to Know X 0 Mean X 27 2 2 0 Standard Deviation S 2 or o l ZCXN u zcx gt02 or 02 zcx oz 0 Variance 2 11 1 N 0 Percentage 29100 n100 0 Proportion p g 0 Z Score discussed for 2415 Z 2 or z 0 Anything else you may learn later that seems important Example 1 If the mean score on a questionnaire about perceived happiness is 5 and the sum of all the scores in the sample is 1000 What is the sample size Step I Choose the proper equation XX X Tl Step 2 Plug in known quantities 1000 5 Tl Step 3 Solve for unknown quantity 5nl 000 Multiply both sides by n to get rid of fraction Divide both sides by 5 to get n by itself Example 2 If s50 and n10 then What is SS Note These questions may be presented in the form of a word problem see above or as a listing of information this problem Make sure you can identify the given information in either format It is also important to decide whether or not you are dealing with a sample or a population as the equations and symbolization may di er If this problem said that 650 and N10 you should know that you are dealing with the population standard deviation and be able to identify the proper equation Given the way the information is actually presented in this problem you will be dealing with the sample standard deviation Your answer would be di erent depending on which equation you choose Step I Choose the proper equation 39 2 s or more usefully s i n 1 n 1 Step 2 Plug in known quantities 50 i 10 1 Step 3 Solve for unknown quantity 2500 57 Square both sides to get rid of radical SS22500 Multiply both sides by 9 to get SS by itself Reporting Results When reporting results mean and standard deviation in a written format use the model M SD Example Students who complete their homework assignments achieve higher test scores M 92 SD6 than those who do not do their homework M 74 SD9 Note Make sure to italicize the mean and standard deviation and include them in parentheses after the relevant claim If you have statistics for multiple groups you may find it easier to present the information in a table Example Did Homework Didn t do Homework Females M 94 M 79 SD 4 SD 9 Males M 91 M 72 SD 6 SD 7 Degrees of Freedom This concept is complicated at best and will be explained more fully at a later date dfThe number of scores in a sample that are independent and free to vary In other words the df refers to how many scores can be randomly chosen while ensuring that the statistic stays the same The df for means is n1 o This means that n1 scores can be randomly chosen but the last value must be specifically chosen in order to make the mean a certain number 0 For example if we have five scores and we want the mean to be 15 four of the five scores can be any random number but we must decide what the fifth number must be in order to ensure that the mean is 15 The first four scores could be 10 5 15 and 20 In order for the mean to be 5 the last score must be 25 If it is any other number the mean won t be 5 This is proven below 0 O O O x 2 Choose proper equation 15 w Plug in known quantities 15 505 Simplzm 75 50 x Multiply both sides by 5 to get rid of fraction o Subtract 50 from both sides to get x by itself Double Check your work 2 w 75 5 15 O 0 Therefore the last score must be 25 If you plug in any other number for the last X value the mean will not equal 15 o The first four scores in the example above were irrelevant it is the last score that matters For example if we still want the mean to be 15 but the scores are now 30 25 5 and 10 the last score would have to be 5 not 25 like above This is proven below 0 O x 2 Choose proper equation 15 w Plug in known quantities 15 705 Simplzm 75 70 x Multiply both sides by 5 to get rid of fraction x 5 I Subtract 70 from both sides to get x by itself Double Check your work 2 w 7 55 15 Therefore the last score must be 5 If you plug in any other number for the last x value the mean will not equal 15 0 This concept may seem foreign but you have already been using it In the equation for standard deviation S 2X gt02 T the n1 1n the denom1nator refers to the degrees of freedom After all as stated above for means df n1 The Normal Curve The normal curve is a bellshaped symmetrical model of scores that represents the probability density function of a normal distribution The Normal Curve Above is a graph of a normal curve As you can see it is symmetrical and centered on the mean Measurements are made in increments of standard deviation and the percentages above re ect the percentage of scores that should fall into each category For instance about 68 of data fall Within one standard deviation on either side of the mean 95 of data fall in the ZSD to ZSD range and 997 of data fall in the 3SD to 3SD range Therefore only a very small percentage of scores Will fall outside of 3 SDs of the mean Skewed Curves F Right Skewed Positively Skewed Distribution rx l lruquumzius Furqtmncies l l l l l l l l l 1 l l i l l l l I Mode Median Mean Left Skewed Negatively Skewed Distribution Mean Median Mode A rightskewed or positively skewed distribution has more data in the left tail than a normal distribution would have The result is a hump at the left of the graph and tail tapering to the right A leftskewed or negatively skewed distribution has more data in the right tail than a normal distribution would have The result is a hump at the right of the graph and tail tapering to the left Note You must be able to remember which graph is rightskewed and which one is leftskewed Think about which side the tail is on A rightskewed distribution has a tail trailing to the right and a leftskewed distribution has a tail trailing to the left A positively skewed rightskewed distribution has a tail trailing toward the higher more positive values to the right of the graph and a negatively skewed leftskewed distribution has a tail trailing toward the lower more negative values to the left of the graph Note Also note where the mean median and mode lie in relation to each other on each of these graphs Remember that the mean median and mode should be equal in a completely normal distribution 2415 Z Scores Location of Scores and Standardized Distributions ZScore Basics Zscores are standardized scores which allow for the comparison of raw scores especially if they are on different scales Zscores serve two main objectives 1 They make raw scores more meaningful o A zscore will tell you the exact location of a raw score in a distribution 2 They can be used to transform and standardize whole distributions 0 Helps us compare results across different tests 0 Makes different distributions equivalent allows comparisons between distributions Zscores consist of a number and a sign where positive scores are above the mean and negative scores are below the mean The number is the distance between the mean and the raw score in units of standard deviation Zdistributions use the normal curve and zscores are in units of standard deviation 0 For example a zscore of 1 would be 1 standard deviation from the mean 0 The mean of the data will always have a zscore of 0 Y A z distribution I l l i i u gt2 3 2 1 o 1 2 3 t l t 1 l l 1 l i i i i i l gt X uSa u ZO ulo p n1a p20 p30 The Translation of X to Z by the Transtormation Z a X pyo As shown by the graph above zdistributions are always normal curves regardless of what the original distribution looked like A zscore of 0 is assigned to the mean of the original distribution u or E A zscore of is assigned to the value of the mean 1 standard deviation 0 or sd u 1 0 A zscore of I is assigned to the value of the mean 1 standard deviation uI o Other zscores 3 2 1 0 I 2 3 etc are determined in a similar fashion Calculating ZScores You calculate the zscores of populations and samples the same way but the symbols used are a little different The zscore equations are 0 Population Z 1 x JE 0 Sample 2 2 5 1 0 The X in the equation is a raw score These equations are most often used to calculate z scores or raw scores depending on What information you are given Example 1 If u40 and 64 What is the zscore of a raw score of 46 Step I Choose the proper equation xu z 039 Step 2 Plug in known quantities 46 40 4 Step 3 Solve for unknown quantity 29 lz15 Example 2 If i20 and sd4 What raw score corresponds to a zscore of 15 Step I Choose the proper equation xi SD Step 2 Plug in known quantities 1 5 x 20 39 4 Step 3 Solve for unknown quantity 6 X20 Multiply both sides by 4 to get rid of fraction X 26 Add 20 to both sides to get x by itself Note You should be able to easily convert between zscores and raw scores and vice versa Pay attention to whether or not the problem is about a population or a sample and know which symbols go with each If a problem does not specifically state whether or not it is referring to a population or a sample you should assume that it refers to a sample 2615 Standardizing and Transforming Distributions ZDistributions If we transfer all scores in a distribution into zscores the zdistribution has 3 properties 0 The zdistribution has the same shape as the distribution of raw scores Scores keep their position 0 All zdistributions have a mean of 0 The original mean is transferred into this value 0 The standard deviation is always 1 Standardized Distributions A standardized distribution is composed of scores that have been transformed to create predetermined values for the mean and standard deviation They can be used to compare scores from dissimilar distributions Transforming a distribution into a completely new standardized distribution takes two main steps 0 Transfer raw scores into zscores 0 Change the zscores into the new standardized scores 0 Standard score new mean z times new standard deviation Note Creating a new standardized distribution involves choosing an arbitrary mean and standard deviation for the new distribution Once the original raw scores are transformed into z scores they can be transformed into new scores for the new standardized distribution using the zscore formula Example 1 A population has a mean of 37 and a standard deviation of 2 If this distribution is transformed into a new distribution with a mean of 100 and standard deviation of 20 what new value will be obtained for a score of 35 Identify the information for both the old and new distributions D1 D2 u37 u100 62 620 x35 X Note DI stands for Distribution 1 the original and D2 stands for Distribution 2 the new Step I 39 Use D1 to calculate a zscore 35 37 Z 2 Plug in known quantities Z l Simplify Step 2 Use the zscore and D2 information to calculate x for new distribution Z 039 1 36 21000 Plug in known quantities 20 XlOO Multiply both sides by 20 to get rid of fraction l X 80 I Add 100 to both sides to get x by itself Note x80 means that a raw score of 35 in the original distribution D1 which has a zscore of I as calculated in Step I is equivalent to a score of 80 in the new distribution D2 Therefore all three of these values fall in the same place on a graph Graph I z score 3 2 1 O 1 2 3 D1 31 33 35 37 39 41 43 D2 40 6O 80 100 120 140 160 Explanation of the Graph The above graph shows the scores for both the old and new distributions at particular z scores When creating one of these graphs the zscores always range from 3 to 3 with 0 being at the mean and the standard deviation being 1 We were told in the problem that the mean of the original distribution was 37 so 37 corresponds to a zscore of 0 and is placed on the graph accordingly We also know that the original distribution D1 has a standard deviation of 2 so it is quite easy to assign values to the remaining zscores A zscore of 1 is one SD above the mean so we simply add 2 the SD to 37 the mean to get the raw score that corresponds to a zscore of 1 We place 39 on the graph accordingly Adding another 2 will give us a score of 41 for a z score of 2 and so on A similar method can be used to determine the scores for the new distribution D2 We were told that the mean of the new distribution is 100 and the standard deviation is 20 A zscore of 1 is one SD above the mean so we simply add 20 the SD to 100 the mean to get the raw score that corresponds to a zscore of 1 We place 120 on the graph accordingly Adding another 20 will give us a score of 140 for a zscore of 2 and so on This basic principle allows you to easily check your work We determined that a score of 35 in D1 corresponds to a score of 80 in D2 Is that correct We know that X 35 has a zscore of 1 in D1 Therefore X in D2 also has a zscore of 1 Knowing that the mean of D2 is 100 and the standard deviation is 20 we can simply subtract 20 from 100 to get 80 Therefore a score of 35 in D1 does indeed become a score of 80 in D2 Checking your work in this way becomes more difficult when you have zscores that are not whole numbers such as z 231 but it is very easy for the basic zscores It is recommended that you graph the diagram above whenever you solve one of these problems although all the information you really need to include is the scores that correspond to the means and the information in the blue box the score in question

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