Social Statistics

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2/8 (Friday)
Standard Deviation cont’d
Example: Amount of money earned by new immigrants
Sample: 12, 15, 16, 20, 25, 36, 40
Step 1: Find the mean
* x̄=164/7
* mean is 23.4
Step 2: Find how much each observation deviates from the mean * 12 - 23.4 = -11.4
* 15 – 23.4 = -8.4
* 16 – 23.4 = -7.4
* 20 – 23.4 = -3.4
* 25 – 23.4 = 1.6
* 36 – 23.4 = 12.6
* 40 – 23.4 = 16.6
* Note: all observations below mean will be negative, all above will be positive Step 3: Square the deviations
* (-11.4)² = 129.96
* (-8.4)² = 70.56
* (-7.4)² = 54.76
* (-3.4)² = 11.56
* (1.6)² = 2.56
* (12.6)² = 158.76
* (16.6)² = 275.56
Step 4: Add them together
* 129.96 + 70.56 + 54.76 + 11.56 + 2.56 + 158.76 + 275.56 = 703.72 Steps 5 & 6:
* Divide sum from sample size
* 703.72/7 = 100.53
* Take square root of the number
* √100.53 = 10.03 – round two places out
* This means that, on average, the income of new immigrants deviates almost $10.000 from the mean. * This is a relatively large deviation (almost half the mean) * There is a lot of variability in how much new immigrants earn. The Normal Range

* Within 1 standard deviation of the mean
* Contains cases considered close to the norm
Variation
* Very similar to standard deviation
* Formula:
* To calculate variation, square the standard deviation
What do we know?
* Distributions – categorizing and graphing frequencies * Central tendency and variability
* Conclusions based on what we observe
* Example: large standard deviations let us conclude distribution has lots of variability * Now, we will test relationships based on probability
Probability – mathematical measure of the likelihood of an even occurring * Chance of the desired even occurring written in %, proportion, or ration. * 40% chance of rain
* Batting average .313
* Probability of a royal flush is 1:649,740
* P(E) = Number of trials in which E occurs / Total number of trials * P(E) is the probability of event (E)
* Probability is equal to the number of times an event occurs divided by the total number of times an event can occur. * Example: 7 female jurors. What is the probability of randomly choosing a female juror. 7/12 Addition Rule

* P(A or B) = P(A) + P(B)
* Probability of A or B happening but not both
* Probability for either event is equal to their separate probabilities combined * Example: Defendant in murder case
* .52 probability of conviction of murder
* .26 probability of conviction of a less charge
* .22 probability of being aquitted
* Only works when categories can’t overlap (when people can’t be in more than one group) * All possible outcomes add up to 1
2/11 HW 4 due Friday but can turn it in Wednesday
Test on Monday!!!!
Test questions are from homework, but changed numbers
Probability and the Normal Curve
Multiplication Rule
* P(A and B) = P(A) x P(B)
* Outcome of both events occurring is the product of both separate probabilities * Assumes independent outcomes
* Occurrence of on outcome doesn’t change the other
* Example: A city police department reports a 60% clearance rate of all cases * What is the probability they would clear 2 cases in combination? * P(Cases A & Cases B cleared) = P(Case A cleared) x P(Case B cleared) * .36 probability of clearing both cases

Probability Distribution
* Derived from probability theory, not from observation
* Specify the possible values of a variable and calculate the probabilities associated with each * Coin Flip
* Probability of getting heads or tails = .50
* What is the probability of getting 2 tails when flipping a coin twice? Coin Example
* We have 3 known outcomes
* Both head, both tails, one of each
* Use addition/multiplication rules:...
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