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:...