1. Descriptive Statistics

Measures of Central Tendency

• Mean

  • Sum of all values divided by the number of values

  • Formula: Mean = Total/Number of Values

• Median

  • Middle value when data is ordered

  • If even number of values, median is the average of the two central values

• Mode

  • Most frequently occurring value in a dataset

Measures of Spread

• Range

  • Difference between the highest and lowest values

  • Formula: $\text{Range} = \text{Max} - \text{Min}$

• Variance

  • Average of the squared differences from the mean

• Standard Deviation

  • Square root of the variance

  • Formula: $\sigma = \sqrt{\frac{\sum{(x - \bar{x})^2}}{n}}$

2. Probability

Basic Concepts

• Probability of an Event

  • Formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Complementary Events

  • Formula: $P(\text{not } E) = 1 - P(E)$

Probability Distributions

• Discrete Probability Distributions

  • Probability mass function (pmf)

  • Example: Rolling a fair die

• Continuous Probability Distributions

  • Probability density function (pdf)

  • Example: Heights of people

3. Data Presentation

Graphs and Charts

• Bar Chart

  • Used for categorical data

• Histogram

  • Used for continuous data divided into bins

• Pie Chart

  • Shows proportions of a whole

• Box Plot

  • Displays distribution based on quartiles

  • Can also reveal skewness

  • Can include/exclude outliers

• Cumulative Frequency Graph

  • Displays points graphed from a table

  • Used to find data of 5-point summary (akin to box plots)

• Scatter Plots

  • Displays relationship (strength & direction) between two variables as graphed data points.

  • Key words for describing scatter plot data include positive, negative, no correlation, strong, moderate, weak.

    • Also make sure to note if there are or not any present possible outliers.

Tables

• Frequency Tables

  • Organize data into categories with frequency counts

• Grouped Frequency Tables

  • Data divided into ranges or intervals

4. Sampling Methods

Types of Sampling

• Random Sampling

  • Every member of the population has an equal chance of being selected

• Systematic Sampling

  • Selecting every nth member from a list

• Stratified Sampling

  • Population divided into subgroups, sample taken from each subgroup

• Cluster Sampling

  • Population divided into random groups (clusters), then the clusters to collect data from are randomly selected

• Quota Sampling

  • Non-probability, balanced method in which the population is divided into groups (quotas) based on categories (i.e., age, gender, etc.) to ensure each quota has an equal size.

Sample vs. Population

• Sample

  • A subset of the population used to estimate characteristics of the whole

• Population

  • The entire group being studied

5. Hypothesis Testing

Steps in Hypothesis Testing

• State Hypotheses

  • Null hypothesis

  • $H_0:$ $\text{No effect or difference}$

  • Alternative hypothesis $\newline H_1: \text{There is an effect or difference}$

• Choose Significance Level

  • Commonly $\alpha = 0.05\ (5\%)$

  • Calculate test statistic and compare with critical value

  • Reject or fail to reject the null hypothesis

• P-Value

  • The probability that the results/observations are 

    • Low p-value (< .05) → strong evidence against null evidence indicating a significant effect of observations

    • High p-value (≥ .05) → weak/insufficient evidence against null hypothesis indicating observations are likely a coincidence/random chance

Errors

• Type I Error

  • A false positive, meaning that you falsely reject a true null hypothesis.

• Type II Error

  • A false negative, meaning that you fail to reject a false null hypothesis.

  • In real life scenarios, Type II Errors commonly result in more serious/dangerous errors than Type I. 

• On the other hand, the two non-error/correct decisions are:

  • Rejecting a false null hypothesis

  • Not rejecting a true null hypothesis. 

• Notes!

  • Never be too affirmative → we never accept a null hypothesis; only reject or not reject.

  • The language of statistics is very precise in the real world as well as on the exam.  

Types of Tests

• t-Test

  • $t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}$

    • $\bar{x}$ = sample mean

    • $\mu$ = population mean

    • ${s}$ = sample std. deviation

    • ${n}$ = sample size

• Chi-Square Test

  • $\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

    • ${O_i}$ = observed frequency in category ${i}$

    • ${E_i}$ = expected frequency in category ${i}$

      • Check Formula Sheet

Summary

• Mean, median, and mode → measures of central tendency.

• Standard deviation → provides insights for data variability.

• Probability rules → help w/ understanding likelihoods.

• Graphs and tables → useful for presenting and interpreting data clearly.

• Sampling methods → used for reliability of data.