DESCRIPTIVE STATISTICS
Categorical Variable: data based on a name or label (e.g. color).
Quantitative Variables: numerical data, a measurable value (e.g. height).
Categorical Variables
Quantitative variables
Bar Graph
Dotplot
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*
*
*
*
*
*
*
Pie Chart
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Stems
Red
150
140
130
120
110
100
90
80
Stem plot
*
*
*
*
*
Leaves
Yellow
1
Orange
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Green
Blue
Indigo
Violet
26
4579
12225799
0234457899
11478
Key: 110 7 represents an IQ score of 117
Box plot
Histogram
Patterns of Distribution:
Resistant: not affected by outliers.
Percentile: the pth percentile of a distribution is the value such that p percent of
observations fall below it.
Mode: most occurring value, signified by a peak on a graph.
 Shape
-Symetric
Unimodal
Bimodal
-Uniform
-Skewed
Left
Right
Center
-Median (50th Percentile): middle of the observations (resistant).
1
-Mean: 𝑛 ∑ 𝑥 (non-resistant).
 Spread
-Range: difference between minimum and maximum value (non-resistant). It is
a single number, not an interval.
-Inner Quartile Range(IQR): Q3 – Q1 (resistant).
-Variance (σ2-population/s2-sample): average of squared distances from the
1
1
mean (non-resistant). 𝜎 2 = 𝑛 ∑(𝑥 − 𝜇)2 and 𝑠 2 = 𝑛−1 ∑(𝑥 − 𝑥̅ )2 .
-Standard Deviation (σ-population/s-sample): square root of variance (non
resistant).
 Outliers
-Outlier on the lower end: 𝑥 < 𝑄1 − 1.5𝐼𝑄𝑅
-Outlier on the upper end: 𝑥 > 𝑄3 − 1.5𝐼𝑄𝑅
Z-score: standardized value of x. 𝑧 =
Normal Distribution

𝑥−𝜇
𝜎
Empirical Rule:
𝑃(|𝑧| < 𝜎)=0.68
𝑃(|𝑧| < 2𝜎)=0.95
𝑃(|𝑧| < 3𝜎)=0.997
Linear Regression
Correlation Coefficient (r): standardized measure of the strength of association of bivariate quantitative linear data. 𝑟 =
𝑦𝑖 −𝑦
̅
1
𝑥𝑖 −𝑥̅
∑( )( )
𝑛−1
𝑠𝑥
𝑠𝑥
−1 ≤ 𝑟 ≤ 1 (-1=perfect negative correlation; 0=absolutely no correlation; 1=perfect positive correlation)
Correlation DOES NOT imply causation!
Coefficient of Determination(R2): the fraction of variability of y-variable that is explained by the linear relationship of yvariable on x-variable.
Least-Squares Regression Line: line that minimizes the sum of the squares of the distances of all the points from the
line. LSRL passes through the point (𝑥,
̅ 𝑦̅).
𝑦̂ = 𝑎 + 𝑏𝑥
𝑠𝑦
𝑏=𝑟
𝑠𝑥
𝑎 = 𝑦̅ − 𝑏𝑥̅
Extrapolation: prediction of values outside of the domain of LSRL. It is often not accurate.
Residual: the difference between the observed value and the predicted value. 𝑒̂𝑖 = 𝑦𝑖 − 𝑦̂𝑖 , ∑𝑛𝑖=1 𝑒̂𝑖 = 0, and 𝑠𝑒 2 =
1
∑ 𝑒2.
𝑛−2
When evaluating LSRL, look at the scatter plot, the residual plot, and R2. High R2 doesn’t imply a linear relationship. If
the residual is not randomly scattered (i.e. there is a clear pattern), reject the linear model.
Cautions about Correlation:
Lurking Variable: a variable that influences the interpretation of a model.
Direct Causation: change in x influences change in y.
Common Response: change in x and y cause by another variable.
Confounding: the effect of x is confounded by a lurking variable.
Experimental Design
Observational Study: observes individuals and measures variables of interest without
imposing a treatment(e.g. sample surveys).
Experiment: imposes a treatment on individuals to observe their responses.
Factor: the explanatory variable. It has at least two levels (specific values).
Experimental Unit: an individual on which the experiment is done (human experimental
units are subjects).
Treatment: specific experimental condtion applied to to a unit.
Control Group: a group that the other factors are compared against (old treatment or
placebo).
Avoiding Bias:
Blind or Double-Blind Experiement
Principles of Experimental Design:
 Control the effects of the lurking variable by comparing treatments to the control
group.
 Randomize the assignment to treatments.
 Replicate each treatment on many units to remove natural variation.
The ONLY way to show causation is with a well-designed, well-controlled experiment.
Experimental Designs:
Matched Pairs: each unit receives both treatments in a random order.
Blocking: units are assigned to blocks based on similar characteristics and then randomly
assigned to treatments.
Sampling
Population: an entire group of individuals.
Census: attempts to obtain information about every individual in the population of
interest.
Sample: a part of the population of interest.
Sampling: studying a part of a population in order to gain information about the whole.
Strata: group of similar individuals.
Sampling Methods:
Voluntary
-consists of individuals who chose themselves to respond.
Response
→biased (people with negative opinions are more likely to
respond).
Convenience
-a sample that chooses inidviduals that are easiest to reach
Sampling
→biased based on time and place
Probability
-sample chosen by chance
Sample
-(SRS, Systematic Random Sample)
Simple Random
-of size n consists of n individuals from a population chosen in
Sample (SRS)
such a way that every set of n individuals.
Stratified
-divides population into strata, an SRS is taken in each stratum,
Random Sample
and then the SRSs are combined to form a full sample.
Multistage
-combines several sampling techniques in the selection of the
Sampling
sample.
Cluster Sample
-individuals are chosen in groups that are similar in location.
Bias: systematic favoring of an outcome.
Response bias: caused by the behavior of the respondent or the interviewer (lying,
question wording, race or sex of the interviewer).
Undercoverage: group(s) from the population being left out from the sample-selection
process.
Nonresponse: an individual chosen for the sample cannot be contacted or does not
cooperate.
Probability
Random phenomenon: individual outcomes are uncertain though there is a regular
distribution of outcomes in a large number of repetitions.
Probability: measure of likelihood of an event.
Event: set of outcomes of a random phenomenon.
Sample Space S: a set of all possible outcomes of a random phenomenon.
Probability Rules:
1. 0 ≤ 𝑃(𝐴) ≤ 1
2. 𝑃(𝑆) = 1
3. Complement (Not A): 𝑃(𝐴𝐶 ) = 1 − 𝑃(𝐴)
4. Intersection (A and B): 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴)
5. Union (A or B): 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
P(A∩B)
6. Conditional Probability (A Given B): P(A|B) =
P(B)
7. Independence: 𝑃(𝐴|𝐵) = 𝑃(𝐴) ⇒ 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵)
8. Disjoint: 𝑃(𝐴 ∩ 𝐵) = 0 ⇒ 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
Sampling with Replacement: P(A) is constant for all successive trials.
Sampling without Replacement: P(A) changes for successive trials.
Random Variables
Random Variable X: variable whose value is based on the outcome of a random event.
Discrete Random Variable: random variable with a finite number of possible values. 𝑃(𝑋 =
𝑥) = 𝑝
Continuous Random Variable: random variable whose sample space is an entire
interval. 𝑃(𝑋 = 𝑥) = 0 ; 𝑃(𝑥1 ≤ 𝑋 ≤ 𝑥2 ) = 𝑝.
Normal Distribution N(μ,σ): normal distribution with mean μ and standard deviation σ.
Expected Value (𝝁𝑿 ): mean of random variable. ∑ 𝑥𝑖 𝑝𝑖
Variance of Random Variable (𝝈𝑿 𝟐 ): ∑(𝑥𝑖 − 𝜇𝑥 )2 𝑝𝑖
Standard Deviation of Random Variable (𝝈𝑿 ): √∑(𝑥𝑖 − 𝜇𝑥 )2 𝑝𝑖
Law of Large Numbers: the greater the sample size, the closer 𝑥̅ will get to 𝜇𝑋 .
Mean Rules
Variance Rules
𝜇𝑋+𝑌 = 𝜇𝑋 + 𝜇𝑌
𝜎 2𝑋±𝑌 = 𝜎 2𝑋 + 𝜎 2 𝑌
𝜇𝑋−𝑌 = 𝜇𝑋 − 𝜇𝑌
𝜎 2 𝑎+𝑏𝑋 = 𝑏 2 𝜎 2𝑋
Standard Deviation Rule
𝜇𝑎+𝑏𝑋 = 𝑎 + 𝑏𝜇𝑋
𝜎𝑋±𝑌 = √𝜎 2𝑋 + 𝜎 2 𝑌
Binomial Model
Bernoulli Process:

A finite sequence of Bernoulli Events.

Each event has two possible outcomes.

Each event is independent.

Each event has the same probability.
Binomial Distribution B(n,p): distribution of the count X of successes with probability p in the binomial setting
with n observations.
Complement of p (q): the probability of a failure. 1-p
𝒏!
Permutation 𝒏𝑷𝒌 : the number of different arrangements where order does matter.
𝒌!
Combination 𝒏𝑪𝒌 : the number of different arrangements where the order does not matter. (𝒏𝒌) =
𝒏!
𝒌!(𝒏−𝒌)!
Binomial Probability: 𝑃(𝑋 = 𝑘) = (𝑛𝑘)𝑝𝑘 (𝑞)𝑛−𝑘
Normal Approximation of Binomial Distribution: Suppose B(n,p), then when np≥10 and nq≥10, the
distribution of X can be approximated by the normal model N(𝑛𝑝, √𝑛𝑝𝑞).
(𝑿 = 𝒏)
𝑷(𝑿 > 𝒏)
𝝁𝑿
𝝈𝟐 𝑿
Geometric Model
Probability of X occurring on nth trial
Probability of X occurring after nth trial
Mean of a Geometric Count
Variance of a Geometric Count
= 𝑝𝑞𝑛−1
= 𝑞𝑛
= 𝑝−1
𝑞
= 2
𝝈𝑿
Standard Deviation of a Geometric Count
=√
𝑝
𝑞
𝑝
Sampling Distributions
Parameter: describes a population
Statistic: describes a sample
Sampling distribution: distribution of values taken by a statistic in all possible samples of the same size from the same
population
Unbiased statistic: the mean of a sampling distribution is equal to the true parameter being estimated
Variability of a statistic: spread of a sampling distribution
Notation
p
̂
𝒑
Population Proportion
Sample Proportion
𝝁𝒙̅
𝝈𝒙̅ 𝟐
Mean of Sampling Distribution
Variance of Sampling Distribution
𝝈𝒙̅
Standard Deviation of Sampling Distribution
=
𝝁𝒑̂
𝝈𝒑̂ 𝟐
Mean of Sampling Distribution of Sample Proportion
Variance of Sampling Distribution of Sample Proportion
=𝑝
𝑝𝑞
=
𝝈𝒑̂
Standard Deviation of Sampling Distribution of Sample Proportion
𝑋
=
𝑛
=𝜇
=
𝜎2
𝑛
𝜎
√𝑛
𝑛
=√
𝑝𝑞
𝑛
{
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒 ≥ 10𝑛
𝑛𝑝 ≥ 10
𝑛𝑞 ≥ 10
Properties of Sampling Distributions:

Sampling distribution of 𝑥̅ is normal if the population distribution is normal.

Central Limit Theorem: The sampling distribution of 𝑥̅ is close to the normal distribution N (μ,

n is sufficiently large (𝑛 ≥ 30), even when the population distribution is not normal.
Variability decreases as sample size increases.
Inference
Significance Test:
Null Hypothesis (𝐇𝟎 ): claim about a parameter that is initially believed to be true.
Alternate Hypothesis (𝐇𝒂 ): claim competing with the null hypothesis.
P-Value: the probability of obtaining the observed test statistic assuming the null
hypothesis is true.
Significance Level (α): the probability threshold beyond which we reject the null
hypothesis
Effect: the difference between true and hypothesized mean.
Power : Increasing sample size, effect size, or significance level will increase
power.
Truth about Population
H0 True
H𝑎 True
Type I Error
Correct Decision
Reject H0
Decision
(P=α)
(P=1-β)
Correct Decision
Type II Error
Accept H0
(P=1-α)
(P=β)
Z Statistic
Conditions
INDEPENDENCE
RANDOMIZATION: SRS
TEN PERCENT: 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒 ≥ 10𝑛
NORMALITY: 𝑛𝑝 ≥ 10 and 𝑛𝑞 ≥ 10
Confidence Interval
𝐳 ∗ : confidence level.
One Proportion Z Interval
𝑝̂ ± 𝑧 ∗ √𝑝̂ 𝑞̂⁄𝑛
Two Proportion Z Interval
𝑝̂1 𝑞̂1 𝑝̂2 𝑞̂2
𝑝̂1 − 𝑝̂2 ± 𝑧 ∗ √
+
𝑛1
𝑛2
Significance Tests
One Proportion Z Test
𝑧=
𝑯𝟎 : 𝑝 = 𝑝0
𝑯𝒂 : 𝑝 > 𝑝0
𝑯𝒂 : 𝑝 < 𝑝0
𝑯𝒂 : 𝑝 ≠ 𝑝0
Two Proportion Z Test
𝑃(𝑍 > 𝑧)
𝑃(𝑍 < 𝑧)
2𝑃(𝑍 > |𝑧|)
𝑝̂1 − 𝑝̂2
𝑧=
√
𝑯𝟎 : 𝑝1
𝑯𝒂 : 𝑝1
𝑯𝒂 : 𝑝1
𝑯𝒂 : 𝑝1
= 𝑝2
> 𝑝2
< 𝑝2
≠ 𝑝2
𝑝̂ − 𝑝0
√𝑝0 𝑞0 ⁄𝑛
𝑝̂1 𝑛1 + 𝑝̂2 𝑛2
𝑛1 + 𝑛2
𝑃(𝑍 > 𝑧)
𝑃(𝑍 < 𝑧)
2𝑃(𝑍 < |𝑧|)
𝜎
√𝑛
) when
Inference
Confidence Interval:
Estimate ± Margin of Error
Confidence Level: the probability that the method will give a correct
answer.
Margin of Error: decreasing the confidence level, deacrasing the standard
deviation, and increasing the sample size will decrease the margin of error.
Significance Test Procedure:
1.
Name the test and state 𝐻0 and 𝐻𝑎 .
2.
Check conditions.
3.
Perform appropriate calculations.
P > α → Accept 𝐻0
P ≤ α → Reject 𝐻0
4.
Conclusion.
T Statistic
Conditions
INDEPENDENCE
RANDOMIZATION: SRS
NORMALITY or 𝑛 ≥ 40
Confidence Interval
𝐭 ∗ : confidence level.
One Sample/Matched Pairs T Interval
𝑠
𝑥̅ ± 𝑡 ∗
√𝑛
Two Sample T Interval
𝑥̅1 − 𝑥̅2 ± 𝑡 ∗ √
𝑠1 2 𝑠2 2
+
𝑛1
𝑛2
Significance Tests
One Proportion/Matched Pairs T Test
𝑥̅ − 𝜇0
𝑡=
𝑠
√𝑛
𝑯𝟎 : 𝜇 = 𝜇0
𝑯𝒂 : 𝜇 > 𝜇0
𝑃(𝑇 > 𝑡)
𝑯𝒂 : 𝜇 < 𝜇0
𝑃(𝑇 < 𝑡)
𝑯𝒂 : 𝜇 ≠ 𝜇0
2𝑃(𝑇 > |𝑡|)
Two Sample T Test
𝑥̅1 − 𝑥̅2
𝑧=
𝑠 2 𝑠 2
√ 1 + 2
𝑛1
𝑛2
𝑯𝟎 : 𝜇1 = 𝜇2
𝑯𝒂 : 𝜇1 > 𝜇2
𝑯𝒂 : 𝜇1 < 𝜇2
𝑯𝒂 : 𝜇1 ≠ 𝜇2
𝑃(𝑇 > 𝑡)
𝑃(𝑇 < 𝑡)
2𝑃(𝑇 > |𝑡|)