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

Step 2: Square each value

Step 3: Sum

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 44 640 5 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 50.56 640 387.2 5 5 Answer = 7.11

Practice Use the raw score formula and find the standard deviation of: 6, 3, 4, 10, 8

31 225 5 5 N = 5 X = 31  X2 = 225

1936 44 225 192.2 2.56 = 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

68 1034 5 5 - 1 N = 5 X = 68  X2 = 1034

1936 44 1034 924.8 5.22 = 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 = Ŝ 2 =

Variance S 2,  2 = Ŝ 2 = - 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 = Ŝ 2 =

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 = Ŝ 2 = - 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

Standard Deviation Want to use Ŝ 452 58 8 8 - 1 -1

Standard Deviation Want to use Ŝ 58 452 8 8 - 1 -1

Standard Deviation Want to use Ŝ 452 420.5 7 -1

Standard Deviation Want to use Ŝ 2.12 -1

Boxplots The boxplot graphically displays three different characteristics of the distribution Extreme scores Interquartile range Median

Boxplot

Boxplot Interquartile range 25th - 75th percentile

Boxplot Extreme Scores

Boxplot Median

Boxplot Skew -- Look at the “whiskers” to determine if the distribution is skewed

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

Neuroticism

Extraversion

Conscientiousness

Which distribution has a positive skew?

Which distribution has a negative skew?

Which distribution is most compact? E A B D

Which distribution has a median close to 25?

Which distribution is most symmetrical?

Which distribution has has the largest range?

Practice Make a box plot using the simple frequency distribution of Satisfaction with Life scores on page 27

Practice Page 67 3.25 3.28

Practice 3.25 Use S “hat” 33.16

Practice 3.28 Use S “hat” A = 1.90 B = 1.55 B is more consistent