Data Analysis

Population

The entire group you want to study.

Sample

A portion of the population.

Parameter

A number that describes a population.

Statistic

A number that describes a sample.

Categorical Variable

A variable whose values represent categories or groups.

Numerical Variable

A variable whose values represent counted or measured quantities.

Discrete Variable

A numerical variable that comes from counting.

Continuous Variable

A numerical variable that comes from measuring.

Nominal Measurement Scale

Categories with no ranking or order.

Ordinal Measurement Scale

Categories that have a ranking or order.

Interval Measurement Scale

A numerical scale with meaningful differences but no true zero.

Ratio Measurement Scale

A numerical scale with meaningful differences and a true zero.

Simple Random Sampling

Every individual has an equal chance of being selected.

Systematic Sampling

Choose a random starting point and then select every $k\text{th}$ individual.

Systematic Sampling Formula

$$k = N \div n$$

Stratified Sampling

Divide the population into groups and randomly sample from each group.

Convenience Sampling

Select people because they are easy or convenient to reach.

Potential Bias in Convenience Sampling

Not everyone has a known chance of being selected, so the results can be biased.

Contingency Table

A table used to organize two or more categorical variables.

Overall Percentage Formula

$$\text{Cell} \div \text{grand total}$$

Row Percentage Formula

$$\text{Cell} \div \text{row total}$$

Column Percentage Formula

$$\text{Cell} \div \text{column total}$$

Best Chart for Comparing Two Categorical Variables

Side-by-side or clustered bar chart.

Calculation of Mean

Add all values and divide by the number of values.

Finding the Median

Put the data in order and find the middle value.

Finding the Mode

Find the value that occurs most often.

Range Formula

$$\text{Maximum} - \text{minimum}$$

Sample Variance Formula

Sum of the squared deviations from the mean $\div (n - 1)$

Calculation of Sample Standard Deviation

Take the square root of the sample variance.

Meaning of a Larger Standard Deviation

The data are more spread out.

Coefficient of Variation (CV)

Measures variation relative to the mean.

Coefficient of Variation (CV) Formula

$$\frac{\text{Standard deviation}}{\text{mean}} \times 100\%$$

Five-Number Summary Components

Minimum, $Q_1$, Median, $Q_3$, Maximum.

Interquartile Range (IQR) Formula

$$Q_3 - Q_1$$

Lower Outlier Fence Formula

$$Q_1 - 1.5 \times \text{IQR}$$

Upper Outlier Fence Formula

$$Q_3 + 1.5 \times \text{IQR}$$

Z-Score Formula

$$\frac{X - \text{mean}}{\text{standard deviation}}$$

Z-Score Extreme Outlier Condition

$Z < -3$ or $Z > +3$

Characteristics of a Right-Skewed Distribution

Has a longer tail on the right, and the mean is typically greater than the median.

Characteristics of a Left-Skewed Distribution

Has a longer tail on the left, and the mean is typically less than the median.

Characteristics of a Symmetrical Distribution

The mean and median are approximately equal.

When to Use the Empirical Rule

When the distribution is bell-shaped or symmetric.

Empirical Rule: Percentage Within $1$ Standard Deviation

About $68\%$

Empirical Rule: Percentage Within $2$ Standard Deviations

About $95\%$

Empirical Rule: Percentage Within $3$ Standard Deviations

About $99.7\%$

When to Use Chebyshev's Rule

When the shape of the distribution is unknown.

Chebyshev's Formula

$$\left(1 - \frac{1}{k^2}\right) \times 100\%$$

Chebyshev's Rule: Percentage Within $2$ Standard Deviations

At least $75\%$

Chebyshev's Rule: Percentage Within $3$ Standard Deviations

At least $88.89\%$

Covariance

Measures the linear relationship between two numerical variables.

Correlation Coefficient ($r$)

Measures the strength and direction of the linear relationship between two numerical variables.

Possible Range of Correlation Coefficient ($r$)

$-1$ to $+1$

Meaning of Correlation Close to $+1$

A strong positive linear relationship.

Meaning of Correlation Close to $-1$

A strong negative linear relationship.

Meaning of Correlation Close to $0$

A weak linear relationship.

Does Correlation Prove Causation?

No. Two variables can be related without one causing the other.