Introduction to Statistics

Population vs. Sample

Understanding the difference between a population and a sample is fundamental in statistics.

• Population: The entire group about which information is desired.

• Sample: A subset of the population that is actually observed or analyzed.

• Census: A study in which the sample is the whole population.

• Statistic: A numerical value calculated from a sample.

• Parameter: A numerical value that describes a characteristic of a population.

Example: If Stone surveys 25 people at a dog adoption event, the population might be all American pet owners, while the sample is the 25 people surveyed.

Descriptive Statistics

Types of Bias in Sampling

Biases can distort the results of a study. Recognize and explain the following types:

Note: Small sample size is not a bias by itself.

Classifying Data

Data can be classified by type and level of measurement:

• Categorical (Qualitative): Data that represent labels or groups (e.g., color, brand).

• Quantitative: Data that can be measured or counted.

  • Discrete: Countable values (e.g., number of pets).

  • Continuous: Measurable values (e.g., weight, height).

Levels of Measurement (NOIR):

1. Nominal: Names/labels only, no order (e.g., rock type).

2. Ordinal: Ordered categories, but differences are not meaningful (e.g., race placement).

3. Interval: Ordered, differences are meaningful, but no true zero (e.g., temperature in °F).

4. Ratio: Ordered, differences are meaningful, and there is a natural zero (e.g., weight, age).

Example: "Ratio because 0 ounces means no whipped cream (natural zero)." 0°F is interval, not ratio, because 0°F does not mean "no heat."

Sampling Methods

Different methods are used to select samples:

Stratified vs. Cluster: Stratified = some from every group; Cluster = everyone from some groups.

Experiment vs. Observation: Experiment applies a treatment; observation does not. Placebo is a fake treatment. Blinding means subjects do not know which treatment they receive; double-blind means neither subjects nor researchers know. Confounding occurs when it is unclear which factor caused the result. Randomization assigns by chance. Replication repeats the experiment on many subjects.

Types of Graphs

Shapes: Uniform = flat; Normal = bell, symmetric; Skewed right = long tail right; Skewed left = long tail left.

Standard Deviation and Related Concepts

• Standard deviation (SD): Measures average distance from the mean. Small SD = values close together; large SD = values spread out.

• Variance: SD squared. Used to avoid negatives canceling out.

• Sample SD uses n − 1: This adjustment (degrees of freedom) gives a better estimate for samples.

• z-score: Number of SDs a value is from the mean. Negative = below mean; positive = above. Unusual if beyond ±2 SDs (for normal data).

Descriptive Statistics: Calculations

Mean (Average)

To find the mean:

1. Add all the numbers.

2. Count how many numbers (n).

3. Divide the sum by n.

Formula:

Example: Data: 4, 8, 6, 10, 2. Mean = (4+8+6+10+2)/5 = 6.

Median

The median is the middle value when data are sorted.

1. Sort the numbers.

2. If odd number of values: median is the middle one.

3. If even: median is the average of the two middle values.

Example: 1, 3, 5, 7, 9 → median = 5. 2, 4, 6, 8 → median = (4+6)/2 = 5.

Quartiles and Interquartile Range (IQR)

1. Sort the data.

2. Find the median.

3. Split into lower and upper halves (exclude median if odd n).

4. Q1 = median of lower half; Q3 = median of upper half.

5. IQR = Q3 − Q1.

Example: Data: 1, 3, 5, 7, 9, 11, 13. Q1 = 3, Q3 = 11, IQR = 8.

Range, Midrange, Mode

• Range: Largest minus smallest value.

• Midrange: (Largest + smallest) / 2.

• Mode: Value that appears most often (can be none or more than one).

Example: Data: 1, 3, 5, 7, 9, 11, 13. Range = 12, Midrange = 7.

Skewness from Numbers

• If mean > median: likely skewed right.

• If mean < median: likely skewed left.

• Outliers pull the mean toward the tail.

Example: Data: 1, 2, 3, 4, 40. Mean = 10, median = 3 → likely skewed right.

Percentile of a Value

1. Count how many values are less than x.

2. Divide by total number of values.

3. Multiply by 100.

Example: Scores: 60, 70, 75, 80, 90, 95. Percentile of 80 = 50th percentile.

z-score

1. Subtract the mean from the value.

2. Divide by the standard deviation.

Formula:

Example: Mean = 70, SD = 10, Score = 85. z = (85-70)/10 = 1.5.

Empirical Rule (68-95-99.7 Rule)

• For bell-shaped (normal) data:

  • Within 1 SD: ≈ 68% of data

  • Within 2 SDs: ≈ 95%

  • Within 3 SDs: ≈ 99.7%

Example: Mean = 50, SD = 15. Between 35 and 65 (±1 SD): ≈ 68%.

Chebyshev's Theorem

Applies to any data shape. For any K > 1:

of the data are within K standard deviations of the mean.

Example: K = 2: At least 75% of data within 2 SDs.

Coefficient of Variation (CV)

Compares spread of different data sets (unitless, as a percent):

Smaller CV = more consistent data.

Boxplot and Outliers

1. Find 5-number summary: min, Q1, median, Q3, max.

2. Draw box from Q1 to Q3, line at median, whiskers to min and max.

3. Outlier test: Lower fence = Q1 − 1.5 × IQR; Upper fence = Q3 + 1.5 × IQR. Values outside are outliers.

Example: Data: 2, 4, 5, 6, 7, 8, 30. Upper fence = 14; 30 is an outlier.

Probability

Counting Rule (Multiplication Principle)

For events in sequence, multiply the number of choices at each stage.

Example: 3 shirts × 4 pants = 12 outfits.

Permutations and Combinations

• Permutation (nPr): Order matters (e.g., president, VP).

• Combination (nCr): Order does not matter (e.g., committee).

Formulas:

Permutation:

Combination:

Example: 10 people, choose 3: Committee = 10C3 = 120; President/VP/Treasurer = 10P3 = 720.

Probability Basics

• Probabilities are between 0 and 1; all probabilities sum to 1.

• "OR" means add; "AND" means multiply; "At least one" means 1 minus the probability of none.

Addition Rule: "A or B"

Example: 40% like pizza, 30% like tacos, 10% like both: 0.40 + 0.30 − 0.10 = 0.60.

Multiplication Rule: "A and B"

For independent events:

For dependent events (without replacement), adjust probabilities accordingly.

Example: Flip heads AND roll a 6: 1/2 × 1/6 = 1/12.

"At Least One" Probability

Probability of at least one occurrence:

Example: 3 coin flips, at least 1 head: 1 − (1/2 × 1/2 × 1/2) = 7/8.

Conditional Probability

Probability of B given A:

Example: Bag: 5 red, 5 green. First pick is red (kept out). P(green second | red first) = 5/9.

Tree Diagrams

Used for multi-stage probability problems. Multiply along branches, add probabilities for desired outcomes.

Example: 60% women (30% tried), 40% men (47% tried): 0.60 × 0.30 + 0.40 × 0.47 = 0.368.

Odds

• Odds against: P(not E) : P(E).

• Payout a : b means probability = b / (a + b).

Example: Roulette: P(5) = 1/38, odds against = 37:1.

Exam-Day Game Plan

• Identify question type: word (concept) or calculation.

• Use tables and steps for word questions; always explain in full sentences.

• For calculations, follow the numbered steps, label answers clearly.

• Always specify sample as number + description; always answer discrete/continuous; ratio requires natural zero; always sort before finding median/quartiles.

• For counting/probability: same job = combination, ranked = permutation.

• If stuck, skip and return; eliminate clearly wrong multiple-choice answers first.