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Formula Compute a standard deviation with the Raw-Score Method • Previously learned the deviation formula – Good to see “what's going on” • Raw score formula – Easier to calculate than the deviation formula – Not as intuitive as the deviation formula • They are algebraically the same!! Raw-Score Formula Note: This is the formula for both and S Step 1: Create a table Coffee X 4 10 22 2 6 X 2 Step 2: Square each value Coffee X 4 10 22 2 6 X 2 16 100 484 4 36 Step 3: Sum Coffee X 4 10 22 2 6 X = 44 X 2 16 100 484 4 36 2 X = 640 Step 4: Plug in values N= 5 X = 44 X2 = 640 Step 4: Plug in values 5 5 N= 5 X = 44 X2 = 640 Step 4: Plug in values 44 5 5 N= 5 X = 44 X2 = 640 Step 4: Plug in values 44 5 640 5 N= 5 X = 44 X2 = 640 Step 5: Solve! 1936 44 640 5 5 Step 5: Solve! 1936 44 640 387.2 5 5 Step 5: Solve! 1936 44 64050.56 387.2 5 5 Answer = 7.11 Practice • Use the raw score formula and find the standard deviation of: 6, 3, 4, 10, 8 X 6 3 4 10 8 X = 31 2 X 36 9 16 100 64 X2 = 225 31 5 225 5 N= 5 X = 31 X2 = 225 2.56 = 1936 44 225 192.2 5 5 Ŝ • What if we want to use a sample standard deviation to estimate the population ? • We need to make one small change to the formula to do this • You need to make the s an “unbiased estimator” Ŝ • To do that you use Ŝ – This provides an estimate of the populations variability Remember Just “ - 1” Ŝ Remember S= Just “ - 1” Ŝ= -1 Why? • The first formula is biased -- its answer tends to be too small • Don’t worry about why -- unless you want too!! Practice! • Below is data from 5 people in this class. What is the estimated standard deviation of all the students in this class? Use the Ŝ raw score formula. • Neuroticism scores 12, 15, 22, 10, 9 X 12 15 22 10 9 X = 68 2 X 144 225 484 100 81 2 X = 1034 68 1034 5 5-1 N= 5 X = 68 X2 = 1034 5.22 = 1936 44 1034 924.8 5 4 Variance • The last step in calculating a standard deviation is to find the square root • The number you are fining the square root of is the variance! 2 = population variance Ŝ 2 = sample variance used to estimate 2 Variance S 2, 2 = Ŝ2= Variance S 2, 2 = Ŝ2= -1 There are 12 different formulas! • Standard Deviation – Deviation Formula , S, Ŝ – Raw Formula , S, Ŝ • Variance – Deviation Formula 2, S 2, Ŝ 2 – Raw Formula 2, S 2, Ŝ 2 Review -- Important Formulas • Standard Deviation -- Deviation Formula = Ŝ= Review -- Important Formulas • Standard Deviation -- Deviation Formula Review -- Important Formulas • Variance -- Deviation Formula 2 = Ŝ2 = Review -- Important Formulas • Variance -- Deviation Formula 2 Review -- Important Formulas • Standard Deviation -- Raw Formula and S = Ŝ= -1 Review -- Important Formulas • Variance -- Raw Formula 2 and S2 = Ŝ2 = -1 How to know which to use • 1) Does the question want a standard deviation or a variance (most of the time standard deviations are used) • 2) Is the group of scores a sample or population? • 3) If it’s a sample, do you want to generalize the findings to a population? Practice • You are interested in how citizens of the US feel about the president. You asked 8 people to rate the president on a 10 point scale. Describe how the country feels about the president -- be sure to report a measure of central tendency and the standard deviation. 8, 4, 9, 10, 6, 5, 7, 9 Central Tendency 8, 4, 9, 10, 6, 5, 7, 9 4, 5, 6, 7, 8, 9, 9, 10 Mean = 7.25 Median = (4.5) = 7.5 Mode = 9 Standard Deviation • Want to use Ŝ Standard Deviation • Want to use Ŝ -1 X 8 4 9 10 6 5 7 9 X2 64 16 81 100 36 25 49 81 = 58 = 452 Standard Deviation • Want to use Ŝ 452 58 8 8 - 1-1 X 8 4 9 10 6 5 7 9 X2 64 16 81 100 36 25 49 81 = 58 = 452 Standard Deviation • Want to use Ŝ 58 452 8 8 - 1-1 X 8 4 9 10 6 5 7 9 X2 64 16 81 100 36 25 49 81 = 58 = 452 Standard Deviation • Want to use Ŝ 452 420.5 7 -1 X 8 4 9 10 6 5 7 9 X2 64 16 81 100 36 25 49 81 = 58 = 452 Standard Deviation • Want to use Ŝ 2.12 -1 X 8 4 9 10 6 5 7 9 X2 64 16 81 100 36 25 49 81 = 58 = 452 Boxplots • The boxplot graphically displays three different characteristics of the distribution – Extreme scores – Interquartile range – Median Boxplot 40 30 20 10 0 N= 100 NEUR Boxplot 40 30 Interquartile range 25th - 75th percentile 20 10 0 N= 100 NEUR Boxplot 40 30 Extreme Scores 20 10 0 N= 100 NEUR Boxplot 40 30 Median 20 10 0 N= 100 NEUR Boxplot 40 Skew -- Look at the “whiskers” to determine if the distribution is skewed 30 20 10 0 N= 100 NEUR Create a boxplot • Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 Create a boxplot • Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 Median = 25th = 75th = Lowest = Highest = Create a boxplot • Create a boxplot with this data set 2, 5, 6, 10, 14, 16, 29, 40, 56, 62, 82, 99 Median = 22.5 25th = 6 75th = 62 Lowest = 2 Highest = 99 120 100 80 60 40 20 0 -20 N= 12 VAR00004 Neuroticism 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 N= 130 130 CNEUR MNEUR Extraversion 5.0 4.5 72 152 4.0 3.5 3.0 2.5 106 120 2.0 N= 130 130 CEXTRA MEXTRA Conscientiousness 5 4 3 2 1 0 N= 128 128 CCON MCON Which distribution has a positive skew? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Which distribution has a negative skew? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Which distribution is most compact? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Which distribution has a median close to 25? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Which distribution is most symmetrical? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Which distribution has has the largest range? 120 C 100 E A 80 B 60 D 40 20 0 -20 N= 9 9 9 9 9 VAR00002 VAR00003 VAR00004 VAR00005 VAR00006 Effect Size Index • On average, males are taller than females. • On average, females are less extraverted than males. • On average, middle children tend to be more rebellious than first born children. Effect Size Estimate • Gives a mathematical way to answer the question: How much more? Effect Size Estimate • Ingredients: Two different means an estimated SD of both samples Effect Size Estimate * Note: This is used to examine differences between MEANS Effect Size Estimate ** When calculating put the bigger mean first (so d will always be positive) • d can range from 0 upward • Small effect • Medium effect • Large effect d = .20 d = .50 d = .80 Heights • Do you think the difference in males and females heights is small, medium, or large? • Women = 64.6 inches Men = 69.8 inches = 2.8 inches 69.8 - 64.6 2.8 69.8 - 64.6 2.8 = 1.86 Rebelliousness • Do you think the difference in rebelliousness between middle and first born children is large? • Middle = 19.45 First = 9.88 = 26.68 19.45 - 9.88 26.68 = .36 The mean difference for rebelliousness is 9.57 The mean difference for height is 5.2 Why does Height have a bigger effect size, but smaller mean difference? Rebelliousness Heights 19.45 - 9.88 = .36 26.68 69.8 - 64.6 2.8 = 1.86 Practice • Make a box plot using the simple frequency distribution of Satisfaction with Life scores on page 27 Practice • 4.15 • 4.18