# Range Variance Standard Deviation VARIABILITY a quantitative measure

§Range §Variance §Standard Deviation

VARIABILITY: a quantitative measure of how much scores vary in a distribution. Describes the degree to which scores spread out or cluster together. Examples: • Waiting for a train (reliability) • Waiting for a friend • The people in our neighborhood(s)

Rare Event Approach 1. Experimenter makes a hypothesis about the frequency distribution of a given population. 2. Collects a sample of data from that population 3. Decides how likely it is that the sample came from the hypothesized distribution

1. Waiting for a train/friend. 2. The people in our neighborhoods 3. Is your car a lemon?

VARIABILITY: a quantitative measure of how much scores vary in a distribution. Describes the degree to which scores spread out or cluster together. 85 85 85 95 95 100 Mean = 85, Median = 85, Mode = 85 70 75 75 85 85 85 Mean = 85, Median = 85, Mode = 85

Height 58 64 70 76 82 Weight 110 140 170 200 230

1. Describes the distribution (spread out or clustered together? ) 2. Estimates distance we expect between scores 3. Provides a sense of how well an individual score will represent the population 4. How much error to expect if you’re using a sample to represent the population

VARIABILITY: a quantitative measure of how much scores vary in a distribution. Describes the degree to which scores spread out or cluster together. 85 85 85 95 95 100 Mean = 85, Median = 85, Mode = 85 70 75 75 85 85 85 Mean = 85, Median = 85, Mode = 85

90 75 86 77 85 72 78 79 94 82 74 93 1) Order the observations 72 74 75 77 78 79 82 85 86 90 93 94 2) Highest Obs. – Lowest Obs. 94 – 72 = 22

§ Susceptible to outliers Entire Sample: Highest = 8. 5 Lowest = 0 Range = 8. 5 Delete zero Score Highest 8. 5 Lowest 4 Range= 4. 5

Insensitive to shape of distribution 10 15 20 25 30 35 40

Standard deviation = the average (standard) distance (deviation) of scores from the mean

DEVIATION: distance from the mean (X-μ) Population 1 Score (X) (X-μ) Dev. Score 2 (2 -6) -4 4 (4 -6) -2 6 (6 -6) 0 8 (8 -6) 2 10 (10 -6) 4 μ = 6 ∑ = 0 Average Deviation = 0 / 5 = 0

Score (X) (X-μ) 4 5 6 7 8 μ = 6 Average Deviation = Dev. Score ∑ =

average squared distance from the mean Population 1 Score (X) (X-μ) Dev. Score 2 (2 -6) -4 4 (4 -6) -2 6 (6 -6) 0 8 (8 -6) 2 10 (10 -6) 4 μ = 6 Average Deviation 2 = 40 / 5 = 8 Dev 2 -42 = 16 -22 = 4 02 = 0 22 = 4 42 = 16 ∑ = 40

average squared distance from the mean (the mean squared deviation) Population 1 Score (X) (X-μ) Dev. Score 2 (2 -6) -4 4 (4 -6) -2 6 (6 -6) 0 8 (8 -6) 2 10 (10 -6) 4 μ = 6 Average Deviation 2 = 40 / 5 = 8 Dev 2 -42 = 16 -22 = 4 02 = 0 22 = 4 42 = 16 ∑ = 40

Score (X) (X-μ) Dev. Score Dev 2 4 (4 -6) -2 5 (5 -6) -1 6 (6 -6) 0 7 (7 -6) 1 8 (8 -6) 2 μ = 6 ∑ = Average Deviation 2 =

Standard deviation = the average (standard) squared distance (deviation) of scores from the mean Population 1: Variance = 8 Standard deviation = √ 8 = 2. 82 Population 2: Variance = 2 Standard deviation = √ 2 = 1. 41

Arabic Letters Sample/Statistic Greek Letters Population/ Parameter = Sum of Squares (SS)

Arabic Letters Sample/Statistic Greek Letters Population/ Parameter = Sum of Squares (SS)

Arabic Letters Sample/Statistic Greek Letters Population/ Parameter

Arabic Letters Sample/Statistic Greek Letters Population/ Parameter

Sum of Squares (SS)

Score (X) 1 6 2 2 0 3 2 0 M = 2 (X-M) 2 (1 -2) 2 (6 -2) 2 (2 -2) 2 (0 -2) 2 (3 -2) 2 (2 -2) 2 (0 -2) 2 (-1)2 42 02 02 (-2)2 12 02 (-2)2 (X-M)2 1 16 0 0 4 1 0 4 Σ(x-M)2 = 26 Step 1: Calculate the mean Mean = 2 Step 2: Calculate sum of squares (x-M)2 = 26 Step 3: Divide by (n-1) Variance = 26 / 7 = 3. 71 Step 4: Take the Square root SD = Variance = 3. 71 = 1. 92

Score (X) 1 6 2 2 0 3 2 0 Σx = 16 (Σx)2 = 256

Score (X) 1 6 2 2 0 3 2 0 Σx = 16 (Σx)2 = 256 12 62 22 22 02 32 22 02 X 2 1 36 4 4 0 9 4 0 Σ(x 2) = 58

Score (X) 1 6 2 2 0 3 2 0 Σx = 16 (Σx)2 = 256 Boom! 12 62 22 22 02 32 22 02 X 2 1 36 4 4 0 9 4 0 Σ(x 2) = 58

Score (X) 8 -2 1 3 5 4 4 1 3 3 M= (X-M)2 (X-M) 2 Σ(x-M)2 1. Calculate Sample Variance 2. Calculate Sample SD

Score (X) 8 -2 1 3 5 4 4 1 3 3 Σx = (Σx)2 = X 2 Σ(x 2) =

üSD should not be much larger than ¼ the range ü SD should not be much smaller than 1/6 range (especially if there are no outliers). üMost observations should be within 3 SDs of mean üFor SD, did you take the square root of the variance?

§ Let’s find the Variance and SD for the two Bedtime variables. § Which variable is more variable? That is, which variable has more variability? § Is this surprising? § Is this comparison meaningful? (Hint: when might comparing variances of two different datasets be invalid?

- Slides: 33