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Unformatted text preview: 4/8/11 Announcements Waitlisted Students: Lateadd once seats open I will sign whatever needs my signature You should have an email from the psych department about the specifics Describing Distribuons Last Time FR O FO BE M RE Describing the Shape of DistribuGons Symmetric Asymmetric Symmetry Shape Central Tendency Skew PosiGve Skew (skewed to the right) NegaGve Skew (skewed to the leM) 1 4/8/11 FR BE OM FO RE FR O BE M FO RE Typical Measures of Central Tendency When to Use Each Measure Mean Median 1. When Data are ordinal 2. When interval/ raGo data is highly skewed or there are extreme scores (outliers) Mode 1. When data are nominal. 2. When most common score is the most meaningful way to describe the data Mean Median Mode 1. When data are interval/raGo scale 2. Use unless there is a substanGal reason not to FR FO BE OM RE FR O FO BE M RE PopulaGon Sample PopulaGon Sample "mew" M
"M" X
"X bar" !Xi = N !Xi X= n 2 4/8/11 This Week Visual Displays of Data Measures of Central Tendency Measures of Variability This Week Variability Variability refers to the degree to which scores in a distribuGon are spread out or clustered together Visual Displays of Data Measures of Central Tendency Measures of Variability How much difference to expect from score to score How well the mean represents the scores on the whole, and how well an individual score would represent the whole 3 4/8/11 Other Examples of distribuGons that are different ? Measures of Variability Measures of Variability Range Standard DeviaGon and Variance Range Standard DeviaGon and Variance 4 4/8/11 Range Difference between the highest and lowest value in your data set Ignore the book's treatment of real limits for the calculaGon of the range X 9 3 2 1 0 9 0 = 9 X 4 13 22 6 2 22 2 = 20 4.1 4.3 = 8.4 5 4/8/11 4.1 4.3 = 8.4 4.1 4.3 = 8.4 1.8 1.7 = 3.5 1.8 1.7 = 3.5 4.1 4.3 = 8.4 4.1 4.3 = 8.4 4.1 4.3 = 8.4 4.1 4.3 = 8.4 6 4/8/11 Two Measures of Variability Two Measures of Variability Range Standard DeviaGon and Variance Range Standard DeviaGon and Variance Standard DeviaGon PopulaGon The standard (typical) amount scores deviate from the mean Sample 7 4/8/11 PopulaGon Sample PopulaGon Sample !
"sigma" !
"sigma" s
"s" PopulaGon Sample !
"sigma" s
"s" PopulaGon Standard DeviaGon 8 4/8/11 Cups of Coffee Consumed Each Day N = 10 Individuals Cups of Coffee Consumed Each Day N = 10 Individuals 2, 4, 3, 2, 2, 3, 1, 0, 2, 1 2, 4, 3, 2, 2, 3, 1, 0, 2, 1 i 5 4 X 2 4 3 2 2 3 1 0 2 1 1 2 3 4 5 6 7 8 9 0 1 2 3 4 f 3 2 1 0 10 Cups of Coffee 9 4/8/11 Standard DeviaGon Standard DeviaGon The standard (typical) amount scores deviate from the mean The standard (typical) amount scores deviate from the mean DeviaGon DeviaGon A deviaGon in staGsGcs is the distance a parGcular score is from the mean deviationi = X i ! 10 4/8/11 i X 2 4 3 2 2 3 1 0 2 1 i X 2 4 3 2 2 3 1 0 2 1 Xi  1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 FR O FO BE M RE CalculaGng the Mean  PopulaGon Formula  i X 2 4 3 2 2 3 1 0 2 1 Xi  1 2 3 !X i = N 4 5 6 7 8 9 10 11 4/8/11 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  = 20 / 10 = 2 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 = 20 / 10 = 2 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 = 20 / 10 = 2 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 Signed distances from the mean 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 12 4/8/11 = 20 / 10 = 2 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 1 2 3 4 5 6 7 8 9 10 FR OM FO BE RE 3 3 +1 +5 Will always, by definiGon, be zero (3) + (3) + (1) + (5) = 0 = 20 / 10 = 2 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 Will always, by definiGon, be zero 1 2 3 4 5 6 7 8 9 10 Xi ! Absolute Value ( X i ! )2
Squared Value Sir Arthur Eddington Sir Ronald Fisher 13 4/8/11 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 (Xi )2 1 ( Xi ! ) 2 2 Squared Value 3 4 5 6 7 8 9 10 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 (Xi )2 02 22 12 02 02 12 12 22 02 12 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 (Xi )2 0 4 1 0 0 1 1 4 0 1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 14 4/8/11 i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 (Xi )2 0 4 1 0 0 1 1 4 0 1 12 1 2 3 4 5 6 7 8 9 10 Sum of the Squared DeviaGons aka sum of squares SS
"S  S" i X 2 4 3 2 2 3 1 0 2 1 20 Xi  0 2 1 0 0 1 1 2 0 1 0 (Xi )2 0 4 1 0 0 1 1 4 0 1 12 SS 1 2 3 4 5 6 7 8 9 10 SS 12 N 10 1.2 15 4/8/11 PopulaGon Variance SS 12 N 10 1.2 Mean Squared DeviaGon Variance = Sum of Squared Deviations Number of Scores PopulaGon Variance !2 = SS N ! = !2 Average squared distance of scores to the mean 16 4/8/11 PopulaGon Variance != ! 2 Standard Deviation = Variance SS ! = N
2 Average squared distance of scores to the mean ( Standard Deviation)2 = Variance PopulaGon Standard DeviaGon SS != N Typical (not technically the average) distance of scores to the mean SS 12 N 10 1.2 Mean Squared DeviaGon 17 4/8/11 SS 12 N 10 1.2 2 SS 12 N 10 1.2 2 SS 12 N 10 1.2 2 SS 12 N 10 1.2 2 18 4/8/11 5 4 f 3 2 1 0 0 1 2 3 4 Cups of Coffee Complete Steps for CalculaGng the PopulaGon Standard DeviaGon 1. 2. 3. 4. 5. Find each deviaGon score (Xi ) Square each deviaGon score (Xi )2 Sum the squared deviaGons (Xi )2 Divide SS by the number of scores SS/N Take the square root of the result (SS/N) PopulaGon Standard DeviaGon != # (X i " ) 2 N 19 4/8/11 CalculaGng the Standard DeviaGon CalculaGng the Standard DeviaGon Populaon CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N 20 4/8/11 Check Your Understanding i X 4 8 3 1 Calculate SS A B C D E 8 18 24 26 30 1 2 3 4 Check Your Understanding i Check Your Understanding i X 4 8 3 1 = 4 X 4 8 3 1 = 4 Calculate SS A B C D E 8 18 24 26 I ran out of Gme! 1 2 3 4 Calculate SS A B C D E 8 18 24 26 I ran out of Gme! 1 2 3 4 21 4/8/11 Complete Steps for CalculaGng the PopulaGon Standard DeviaGon 1. 2. 3. 4. 5. Find each deviaGon score (Xi ) Square each deviaGon score (Xi )2 Sum the squared deviaGons (Xi )2 Divide SS by the number of scores SS/N Take the square root of the result (SS/N) 1. Find each deviaGon score i X 4 8 3 1 di = 16/4 = 4 1 2 3 4 1. Find each deviaGon score i 2. Square each deviaGon score i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 = 16/4 = 4 X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 = 16/4 = 4 1 2 3 4 1 2 3 4 22 4/8/11 3. Sum the squared deviaGons i Complete Steps for CalculaGng the PopulaGon Standard DeviaGon 1. 2. 3. 4. 5. Find each deviaGon score (Xi ) Square each deviaGon score (Xi )2 Sum the squared deviaGons (Xi )2 Divide SS by the number of scores SS/N
Take the square root of the result (SS/N) X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 = 16/4 = 4 1 2 3 4 (Xi )2 = 26 Complete Steps for CalculaGng the PopulaGon Standard DeviaGon 1. 2. 3. 4. 5. Find each deviaGon score (Xi ) Square each deviaGon score (Xi )2 Sum the squared deviaGons (Xi )2 Divide SS by the number of scores SS/N Take the square root of the result (SS/N) SS 26 != = = 2.55 N 4 23 4/8/11 i X 4 8 3 1 1 2 3 4 5 6 7 8 1 2 3 4 = 2.55 PopulaGon SS "Computaonal Formula" i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 (X )2 = 26 = 16/4 = 4 1 2 3 4 ( !X i ) SS = !X 2 "
i 2 N 24 4/8/11 SS = !X i2 " ( !X i )
N 2 SS = !X i2 " ( !X i )
N 2 i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 (X )2 = 26 = 16/4 = 4 i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 (X )2 = 26 = 16/4 = 4 1 2 3 4 1 2 3 4 SS = !X i2 " ( !X i )
N 2 SS = !X i2 " ( !X i )
N 2 i X 4 8 3 1 X2 16 64 9 1 i X 4 8 3 1 (Xi)2 = 162 = 256 X2 16 64 9 1 1 2 3 4 1 2 3 4 25 4/8/11 SS = !X i2 " ( !X i )
N 2 SS = !X i2 " ( !X i )
N 2 i X 4 8 3 1 X2 16 64 9 1 i X 4 8 3 1 X2 16 64 9 1 1 2 3 4 1 2 3 4 (Xi)2 = 162 (Xi2)= 90 = 256 (Xi)2 = 162 (Xi2)= 90 = 256 SS = !X i2 " ( !X i )
N 2 i X 4 8 3 1 X2 16 64 9 1 1 2 3 4 SS = 90 ! 256 = 26 4 (Xi)2 = 162 (Xi2)= 90 = 256 26 4/8/11 i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 = 16/4 = 4 1 2 3 4 != SS 26 = = 2.55 N 4 (Xi )2 = 26 PopulaGon Sum of Squared DeviaGons Computaonal Formula PopulaGon Sum of Squared DeviaGons Computaonal Formula !X 2 i ( !X i ) "
N 2 Definional Formula !(X i " ) 2 !X 2 i ( !X i ) "
N 2 Definional Formula !(X i " ) 2 27 4/8/11 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 Check Your Understanding Suppose = 8, SS = 16, and N = 4 Calculate A B C D E 8 16 2 4 I ran out of Gme! SS != N Check Your Understanding Suppose = 8, SS = 16, and N = 4 Calculate A B C D E 8 16 2 4 I ran out of Gme! Suppose = 8, SS = 16, and N = 4 Calculate 28 4/8/11 Suppose = 8, SS = 16, and N = 4 Calculate Suppose = 8, SS = 16, and N = 4 Calculate Suppose = 8, SS = 16, and N = 4 Calculate PopulaGon Sample !
"sigma" s
"s" 29 4/8/11 PopulaGon Sample !
"sigma" s
"s" Sample Standard DeviaGon Standard DeviaGon Sample Standard DeviaGon An esGmate based on sample data of the standard deviaon of the populaon from which the sample was drawn The standard (typical) amount scores deviate from the mean 30 4/8/11 Parameters and StaGsGcs Parameters and StaGsGcs PopulaGon Sample PopulaGon Sample X s X s ? ? STATISTICS STATISTICS PARAMETERS PARAMETERS Parameters and StaGsGcs Parameters and StaGsGcs PopulaGon Sample PopulaGon Sample X s ? ? X s ? ? STATISTICS STATISTICS PARAMETERS PARAMETERS 31 4/8/11 Parameters and StaGsGcs Parameters and StaGsGcs PopulaGon Sample PopulaGon Sample ? ? A ESTIM TION X s STATISTICS ? ? A ESTIM TION X s STATISTICS (ESTIMATES) PARAMETERS PARAMETERS EsGmaGon The process of esGmaGng the values of parameters based on measured/empirical data that has a random component. FR O FO BE M RE Sampling Error The discrepancy (or amount of error) that exists between a sample staGsGc and the corresponding populaGon parameter 32 4/8/11 EsGmaGon The process of esGmaGng the values of parameters based on measured/empirical data that has a random component. ProperGes of EsGmators ProperGes of EsGmators Consistency: "Does it get berer with more data?" ProperGes of EsGmators Consistency: "Does it get berer with more data?" (RelaGve) Efficiency: "Does it err less than other esGmators?" 33 4/8/11 ProperGes of EsGmators Consistency: "Does it get berer with more data?" (RelaGve) Efficiency: "Does it err less than other esGmators?" Sufficiency: "Does it use all the data?" ProperGes of EsGmators Consistency: "Does it get berer with more data?" (RelaGve) Efficiency: "Does it err less than other esGmators?" Sufficiency: "Does it use all the data?" Bias: "Does it over or underesGmate the true value on average?" Biased EsGmator An esGmator is said to be biased if on average it overesGmates or underesGmates its corresponding populaGon parameter Unbiased EsGmator An esGmator is said to be unbiased if on average it is expected to equal its corresponding populaGon parameter 34 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 SAMPLE N = 50 X = 101.1 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 SAMPLE N = 50 X = 101.1 " " = 12 SAMPLE N = 50 X = 101.1 " " = 12 35 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 SAMPLE N = 50 X = 101.1 " " = 12 SAMPLE N = 50 X = 101.1 " " = 12 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 SAMPLE N = 50 X = 101.1 " " = 12 SAMPLE N = 20 X = 101.1 " " = 12 36 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 != SS N != 9018 = 13.43 50 37 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Average POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Average = 126.5 / 10 Average = 126.5 / 10 = 12.65 38 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Biased Esmator on average the sample esGmate does not equal the populaGon parameter Average = 126.5 / 10 = 12.65 Average = 126.5 / 10 = 12.65 POPULATION N = 20,000 = 100 = 15 If we calculate the standard deviaGon of a sample
using the formula for , and then use it to infer about the standard deviaGon of the populaGon the sample is from, we will tend to underesGmate the true value of Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Average = 126.5 / 10 = 12.65 39 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Average = 126.5 / 10 = 12.65 Average = 126.5 / 10 = 12.65 Sample Standard DeviaGon corrected for the purpose of esmaon 40 4/8/11 CalculaGng the Standard DeviaGon Populaon Sample Standard DeviaGon corrected for the purpose of esmaon to correct for underesmaon SS = !(Xi " )2 != SS N CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N != SS N Sample Sample SS = !(Xi " X)2 41 4/8/11 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N != SS N Sample SS = !(Xi " X)2 Sample SS = !(Xi " X)2 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N != SS N Sample SS = !(Xi " X)2 Sample s= SS n !1 SS = !(Xi " X)2 s= SS n !1 42 4/8/11 POPULATION N = 20,000 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 = 100 = 15 != SS N Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Sample SS = !(Xi " X)2 s= SS n !1 Average = 126.5 / 10 = 12.65 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65
Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Sample 05: s = 16.75 Sample 06: s = 14.39 Sample 07: s = 15.66 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Average = 126.5 / 10 = 12.65 s= SS n !1 Average = 126.5 / 10 = 12.65 43 4/8/11 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Sample 05: s = 16.75 Sample 06: s = 14.39 Sample 07: s = 15.66 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 Sample 06: s = 14.39 Sample 07: s = 15.66 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Average = 126.5 / 10 = 12.65 Average = 126.5 / 10 = 12.65 POPULATION N = 20,000 = 100 = 15 POPULATION N = 20,000 = 100 = 15 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 = 151.83 / 10 Sample 06: s = 14.39 Sample 07: s = 15.66 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 = 151.83 / 10 Sample 06: s = 14.39 Sample 07: s = 15.66 = 15.18 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 Average = 126.5 / 10 = 12.65 Average = 126.5 / 10 = 12.65 44 4/8/11 POPULATION N = 20,000 = 100 = 15 s 2 Standard deviaGons for 10 samples calculated using PopulaGon Formula (formula for ) Sample 01: = 13.43 Sample 02: = 11.86 Sample 03: = 11.87 Sample 04: = 13.12 Sample 05: = 13.96 Sample 06: = 11.97 Sample 07: = 13.05 Sample 08: = 9.65 Sample 09: = 15.25 Sample 10: = 12.34 Standard deviaGons for 10 samples calculated using corrected Formula (formula for s) Sample 01: s = 16.12 Sample 02: s = 14.23 Sample 03: s = 14.24 Sample 04: s = 15.75 Average s Sample 05: s = 16.75 = 151.83 / 10 Sample 06: s = 14.39 Sample 07: s = 15.66 = 15.18 Sample 08: s = 11.58 Sample 09: s = 18.30 Sample 10: s = 14.81 2 Average = 126.5 / 10 = 12.65 s 2 2 is an unbiased esmator of: s 2 X 2 45 4/8/11 i X 4 8 3 1 1 2 3 4 Sample i CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 X 4 8 3 1 1 2 3 4 != SS N Sample SS = !(Xi " X)2 s= SS n !1 46 4/8/11 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 !(X i " X ) 2
i X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 X = 16/4 X = 4 SS != N 1 2 3 4 Sample SS = !(Xi " X)2 (Xi X)2 = 26 s= SS n !1 !(X i " X ) 2
i CalculaGng the Standard DeviaGon X 4 8 3 1 di 44 = 0 84 = 4 34 = 1 14 = 3 di2 02=0 42=16 12=1 32=9 X = 16/4 X = 4 Populaon SS = !(Xi " )2 1 2 3 4 != SS N (Xi X)2 = 26 Sample SS = !(Xi " X)2 s= SS n !1 47 4/8/11 CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N s= SS = n !1 26 = 2.94 4 !1 Sample SS = !(Xi " X)2 s= SS n !1 Sample i Sample i X 4 8 3 1 1 2 3 4 5 6 7 8 X 4 8 3 1 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 s = 2.94 s = 2.94 esmate of populaon = 2.94 48 4/8/11 Parameters and StaGsGcs Sample i X 4 8 3 1 1 2 3 4 5 6 7 8 1 2 PopulaGon Sample 3 4 ? ? A ESTIM TION X s STATISTICS (ESTIMATES) s = 2.94 esmate of populaon = 2.94 PARAMETERS CalculaGng the Standard DeviaGon Populaon SS = !(Xi " )2 != SS N Sample SS = !(Xi " X)2 s= SS n !1 49 4/8/11 = 1.0 This Week = 0.5 Visual Displays of Data Measures of Central Tendency Measures of Variability This Week Read For Next Time Ch 5: zScores: LocaGon of Scores and Standardized DistribuGons (Re)read Ch. 4: Variability Visual Displays of Data Measures of Central Tendency Measures of Variability Do Homework 2 (due Monday by 11:00am) Start making note sheets for exam (Two 8.5" x 11" pages, both sides, any format) 50 ...
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This note was uploaded on 01/10/2012 for the course PSYC PSYC 60 taught by Professor ? during the Winter '09 term at UCSD.
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