MGCR 271 Crib Sheet

MGCR 271 Crib Sheet By Kareem Halabi Stem plot: Stem is first few digits, 2nd column represents the last digit of each data point (can be multiple of the same number) 1

Mean (population mean is μ): 𝑥 = ∑ 𝑥𝑖 𝑛

Sample standard deviation (quantifiable spread): 𝑠𝑥 , 𝑠𝑥, 𝜎 ′ or 𝑥𝜎𝑛−1 = √

∑(𝑥−𝑥)2

∑(𝑥−𝜇)2

𝑥−𝑥 𝑠𝑥

(if |𝑧| > 3 then the number

is an outlier). All the z scores of a data set will have 𝑥 = 0, 𝑠𝑥 = 1 Coefficient of Variation: 𝐶𝑉 =

𝑠𝑥 𝑥̅

Intuitive definition of Percentiles: If n data are arranged in numerical order, a number x is called a pth percentile if At least p% of the data ≤ x At least (100-p)% of the data ≥ x

Linear interpolation for percentiles: 1. 2. 3.

𝑛(∑ 𝑥𝑦)−(∑ 𝑥)(∑ 𝑦) 𝑛(∑ 𝑥 2 )−(∑ 𝑥)

Put the data into numerical order Calculate 𝑝% (𝑛 + 1) = 𝑘 (an integer) + 𝑑 (decimal) The pth percentile is: 𝒙𝒌 + 𝒅(𝒙𝒌+𝟏 − 𝒙𝒌 )

First Quartile: Q1 = 25th percentile Median: M = 50th percentile Third Quartile: Q3 = 75th percentile Interquartile Range: IQR = Q3 - Q1 Outliers by boxplot criterion: High outlier: 𝑥 > 𝑄3 + 1.5 × 𝐼𝑄𝑅 Low outlier: 𝑥 < 𝑄1 − 1.5 × 𝐼𝑄𝑅

Q1

M

Q3

Highest non-outlier

𝑛

Negative (left) skew: 𝑥𝑀

𝑃(𝐸 | 𝐹) =

1.

𝑎=

Conditional Probability: Probability of E happening if F also happens

Example Turkey’s boxplot:

Lowest non-outlier

2

𝑛

Z-score: 𝑧 =

1. 2.

Random variable: assigns a numerical value to every possible outcome of an experiment Discrete random variable: When there is a gap between successive possible values (Ex: can have 72 or 73 people but not 72.3) Residual (vertical distance) for (𝒙𝒊 , 𝒚𝒊 ): 𝑒𝑖 = Continuous random variable: can assume all 𝑦𝑖 − 𝑎 − 𝑏𝑥𝑖 vales in some interval Ordinary Least Squares regression: Goal is Probability Distribution Function (PDF): the to minimize ∑ 𝑒𝑖 2 set of all possible values of a discrete random 𝑦 = 𝑎 + 𝑏𝑥 variable together with their probabilities 𝑏=

Median Third Quartile Highest non-outlier (by boxplot criterion)

𝑛−1

Population standard deviation (always use this when dealing with decimal percentages): 𝜎𝑥 = √

3. 4. 5.

Lowest non-outlier (by boxplot criterion) First Quartile

Empirical Rule: If a data set is unimodal and not very skewed, then 1. ~68% of data are within 1𝑠𝑥 of 𝑥̅ 2. ~95% of data are within 2𝑠𝑥 of 𝑥̅ 3. ~99.7% of data are within 3𝑠𝑥 of 𝑥̅

Expected Value of x: 𝐸(𝑥) = 𝜇(𝑥) = 𝜇𝑥 = 𝑥̅ Statistical Independence: The probability of A happening is independent of whether or not B happens (the second flip of a coin is independent from the first flip) 1. 2. 3.

𝑃(𝐴 | 𝐵) = 𝑃(𝐴 | 𝐵̅ ) 𝑃(𝐴 | 𝐵) = 𝑃(𝐴) 𝑷(𝑨 ∩ 𝑩) = 𝑷(𝑨)𝑷(𝑩) (Can be used to determine whether A & B are independent)

Binomial Setup: 1. 2.

There are only two outcomes to an experiment: Success and failure Let p be the probability of success, and it is the same every time

If the experiment is performed n times, the probability of x successes and n-x failures is: 𝒏 ( ) 𝒑𝒙 (𝟏 − 𝒑)𝒏−𝒙 on calc: (nCx)(p^x)(1-p)^(n-x) 𝒙 For Binomial problems: 𝐸(𝑥) = 𝑛𝑝

𝜎(𝑥) = √𝑛𝑝(1 − 𝑝) End of Midterm Material