Ch12: Data-Based and Statistical Reasoning

Measures of Central Tendency

Describe “middle” of data

Outlier

Extremely large or small value compared to other data

Median and mean far apart

Outlier Causes

1. Statistical anomaly
2. Measurement error
3. Distribution not approximated to normal distribution

Mean/Average

Use when values are close together

Outliers shift mean

Median

Midpoint of data in ascending order (50% above and 50% below point)

Use when outliers present

Mode

Value appearing most in data

Normal Distribution

Symmetrical

All measures of central tendency equal

Standard Distribution (Bell Curve)

Transformed normal distribution

Mean = 0 and SD = 1

Skewed Distribution

Asymmetrical (tail on one side)

Different mean, median, mode values

Negatively Skewed Distribution

Tail on left side

Mean < Median

Positively Skewed Distriution

Tail on right side

Mean > Median

Bimodal Distribution

2 peaks

Range

Diff between largest and smallest values

Heavy outlier influence

Interquartile Range (IQR)

Diff between third and first quartile

Determine outliers

IQR: Quartiles

Relative to median

Divide data into groups containing 1/4 of set

IQR: Q1

n x 1/4

Whole Number: Mean of value at position and next highest position

Decimal: Round up

IQR: Q2

Median

IQR: Q3

n x 3/4

Whole Number: Mean of value at position and next highest position

Decimal: Round up

IQR: Outliers

Values 1.5 x IQR below Q1 or above Q3

Standard Deviation (SD)

Measure variability around mean

1/4 of range

SD: Outliers

Values 3 x SD away from mean

SD: Normal Distribution

68% data in 1 SD of mean

95% data in 2 SD of mean

99% data in 3 SD of mean

Independent Events

No effect on each other

Probabilities never change

Dependent Events

Affect each other

Probabilities change depending on prior event outcome

Mutually Exclusive Outcomes

Cannot occur together

Non-Mutually Exclusive Outcomes

Can occur together

Exhaustive Outcomes

Only possible outcomes of event

Hypothesis Testing

Test difference between 2 pop with known distribution

Null Hypothesis

2 pop equal (no difference)

Alternative Hypothesis

Nondirectional: Pop not equal

Directional: Pop A diff in specific way from pop B

P-Value

Test statistic to determine if results obtained by random chance

Significant Level (α)

Compare to p-value

Usually 0.05

p-value > α

Fail to reject null hypothesis

No statistically significant diff between pops

p-value < α

Reject null hypothesis

Statistically significant diff between pops

Type I Error

Report diff between groups when no diff exists (reject null when null true)

α

Type II Error

Report no diff between groups when diff exists (fail to reject null when null false)

β

Power

Reject null when null false

1-β

Confidence

Fail to reject null when null true

1-α

Confidence Intervals

Range of confidence for mean

Sample mean ± SD x z-score

Charts

Visual representation of info

Charts: Pie/Circle

Relative amounts for categorical data

Usually for demographics

Charts: Bar

Categorical data

Bar proportional to value

Charts: Histogram

Numerical data

Determine mode from data distribution

Charts: Box Plot

Numerical data

Boxes: Q1 and Q3

Middle Lines: Median

Whiskers: 1.5 x IQR

Points: Outliers

Charts: Maps

Geographic representation

Graphs

Use axes to provide numerical data

Graphs: Linear

Relationship between 2 variables

Can be linear, parabolic, exponential, logarithmic

Graphs: Semilog

Log graph to linear with axis ratio change (10 exponents)

1 log axis and 1 linear

Graphs: Log-Log

Both axis use constant ratio change

2 log axes

Correlation

Connection/relationship between data

Positive Correleation

Var trend together (both increase)

Negative Correlation

Var trend in opposite dir (1 increase, 1 decrease)

Correlation Coefficient

Between -1 and +1

Show relationship strength

Causation

1 var cause changes in other

Can only be proven with Hill’s criteria