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Statistics Examples & Problems

Master statistics through real-world examples with step-by-step solutions. Each example includes the scenario, data, solution process, and final answer.

Central Tendency

Calculating Class Average

Scenario

A teacher wants to find the average test score for a class of 25 students.

Given Data

Scores: 78, 85, 92, 88, 76, 95, 82, 79, 91, 87, 83, 90, 86, 84, 89, 77, 93, 81, 88, 85, 80, 94, 87, 82, 86

Solution

  1. 1Sum all scores: 78 + 85 + ... + 86 = 2,138
  2. 2Count the number of scores: n = 25
  3. 3Calculate mean: 2,138 ÷ 25 = 85.52

Answer

The class average is 85.52

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Finding the Median Salary

Scenario

An HR manager needs to find the median salary in a department.

Given Data

Salaries: $45,000, $52,000, $48,000, $95,000, $51,000, $49,000, $47,000

Solution

  1. 1Sort the data: $45,000, $47,000, $48,000, $49,000, $51,000, $52,000, $95,000
  2. 2Find the middle position: (7 + 1) ÷ 2 = 4th position
  3. 3The 4th value is the median

Answer

The median salary is $49,000 (note: the outlier $95,000 doesn't affect the median)

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Determining the Mode of Survey Responses

Scenario

A marketing team surveys preferred product colors.

Given Data

Responses: Blue, Red, Blue, Green, Blue, Red, Yellow, Blue, Green, Red, Blue

Solution

  1. 1Count each color: Blue = 5, Red = 3, Green = 2, Yellow = 1
  2. 2Find the most frequent response

Answer

The mode is Blue (appears 5 times)

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Variability & Spread

Quality Control Variation

Scenario

A factory measures the weight of product samples to ensure consistency.

Given Data

Weights (grams): 502, 498, 501, 499, 503, 497, 500, 502, 498, 501

Solution

  1. 1Calculate mean: (502 + 498 + ... + 501) ÷ 10 = 500.1g
  2. 2Find squared deviations: (502-500.1)² + (498-500.1)² + ...
  3. 3Sum of squared deviations: 34.9
  4. 4Sample variance: 34.9 ÷ (10-1) = 3.88
  5. 5Standard deviation: √3.88 = 1.97g

Answer

Standard deviation is 1.97g, indicating consistent product weights

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Comparing Investment Volatility

Scenario

An investor compares the volatility of two stocks over 5 days.

Given Data

Stock A returns: 2%, -1%, 3%, 1%, 0% | Stock B returns: 5%, -4%, 6%, -3%, 1%

Solution

  1. 1Stock A: Mean = 1%, Std Dev = 1.58%
  2. 2Stock B: Mean = 1%, Std Dev = 4.18%
  3. 3Both have the same average return, but different risk levels

Answer

Stock B is more volatile (higher std dev) despite same average return

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Probability

Card Drawing Probability

Scenario

What is the probability of drawing 2 aces from a standard deck without replacement?

Given Data

52 cards total, 4 aces in deck

Solution

  1. 1First ace: P = 4/52
  2. 2Second ace (given first was ace): P = 3/51
  3. 3Combined probability: (4/52) × (3/51)

Answer

P(2 aces) = 12/2652 = 1/221 ≈ 0.45%

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Binomial Probability in Quality Testing

Scenario

A batch has 5% defect rate. What's the probability of finding exactly 2 defects in 20 items?

Given Data

n = 20 trials, p = 0.05, k = 2 successes

Solution

  1. 1Use binomial formula: P(X=k) = C(n,k) × p^k × (1-p)^(n-k)
  2. 2C(20,2) = 190
  3. 3P(X=2) = 190 × (0.05)² × (0.95)^18

Answer

P(exactly 2 defects) ≈ 18.9%

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Committee Selection

Scenario

How many ways can you select a 3-person committee from 10 people?

Given Data

n = 10 people, r = 3 positions (order doesn't matter)

Solution

  1. 1Use combinations formula: C(n,r) = n! / (r!(n-r)!)
  2. 2C(10,3) = 10! / (3! × 7!)
  3. 3= (10 × 9 × 8) / (3 × 2 × 1)

Answer

There are 120 possible committees

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Z-Scores & Normal Distribution

Test Score Percentile

Scenario

A student scores 720 on a test where mean = 650 and std dev = 50. What percentile is this?

Given Data

x = 720, μ = 650, σ = 50

Solution

  1. 1Calculate z-score: z = (x - μ) / σ
  2. 2z = (720 - 650) / 50 = 1.4
  3. 3Look up z = 1.4 in standard normal table

Answer

z = 1.4 corresponds to the 91.92 percentile

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Finding Cutoff Score

Scenario

A company wants to hire top 10% of test takers. Test mean = 500, std dev = 100. What's the minimum score?

Given Data

μ = 500, σ = 100, want P(X > x) = 0.10

Solution

  1. 1Find z for top 10%: z = 1.28 (from z-table)
  2. 2Use z-score formula: x = μ + z × σ
  3. 3x = 500 + 1.28 × 100

Answer

Minimum score needed is 628

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Correlation & Regression

Sales and Advertising Relationship

Scenario

A company wants to understand if advertising spending affects sales.

Given Data

Ad spending ($1000s): 2, 4, 6, 8, 10 | Sales ($1000s): 50, 70, 85, 100, 120

Solution

  1. 1Calculate correlation coefficient using Pearson formula
  2. 2r = Σ[(xi-x̄)(yi-ȳ)] / √[Σ(xi-x̄)²Σ(yi-ȳ)²]
  3. 3r ≈ 0.997

Answer

r = 0.997 indicates very strong positive correlation

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Predicting House Prices

Scenario

Create a model to predict house prices based on square footage.

Given Data

Size (sqft): 1200, 1500, 1800, 2000, 2400 | Price ($1000s): 180, 220, 260, 290, 340

Solution

  1. 1Calculate slope: b = Σ[(xi-x̄)(yi-ȳ)] / Σ(xi-x̄)²
  2. 2b ≈ 0.133 (price increases $133 per sqft)
  3. 3Calculate intercept: a = ȳ - b×x̄
  4. 4a ≈ 22.67
  5. 5Equation: Price = 22.67 + 0.133 × Size

Answer

Regression equation: Price = 22.67 + 0.133 × Size. For 2200 sqft: $315,270

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Confidence Intervals

Estimating Average Customer Wait Time

Scenario

A bank surveys 50 customers and finds average wait time of 8.5 minutes with std dev 2.3 minutes.

Given Data

n = 50, x̄ = 8.5 min, s = 2.3 min, 95% confidence

Solution

  1. 1Standard error: SE = s/√n = 2.3/√50 = 0.325
  2. 2For 95% CI, z* = 1.96
  3. 3Margin of error: E = 1.96 × 0.325 = 0.637
  4. 4CI = 8.5 ± 0.637

Answer

95% CI: [7.86, 9.14] minutes

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Election Poll Analysis

Scenario

A poll of 1,000 voters shows 52% support for Candidate A.

Given Data

n = 1000, p̂ = 0.52, 95% confidence

Solution

  1. 1Standard error for proportion: SE = √[p̂(1-p̂)/n]
  2. 2SE = √[0.52 × 0.48 / 1000] = 0.0158
  3. 3Margin of error: E = 1.96 × 0.0158 = 0.031
  4. 4CI = 0.52 ± 0.031

Answer

95% CI: [48.9%, 55.1%] - race is too close to call!

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Hypothesis Testing

Chi-Square Test for Independence

Scenario

Is there a relationship between gender and product preference?

Given Data

Observed: Men prefer A(30), B(20). Women prefer A(25), B(25). Expected if independent: A(27.5 each), B(22.5 each)

Solution

  1. 1χ² = Σ[(O-E)²/E]
  2. 2= (30-27.5)²/27.5 + (20-22.5)²/22.5 + (25-27.5)²/27.5 + (25-22.5)²/22.5
  3. 3= 0.227 + 0.278 + 0.227 + 0.278 = 1.01
  4. 4df = (rows-1)(cols-1) = 1
  5. 5Critical value at α=0.05: 3.84

Answer

χ² = 1.01 < 3.84, fail to reject H₀. No significant relationship found.

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ANOVA: Comparing Teaching Methods

Scenario

Compare test scores across 3 different teaching methods.

Given Data

Method A: 85, 82, 88, 86 | Method B: 78, 80, 76, 82 | Method C: 90, 92, 88, 94

Solution

  1. 1Calculate group means: A=85.25, B=79, C=91
  2. 2Grand mean: 85.08
  3. 3SSB (between groups) = Σnᵢ(x̄ᵢ - x̄)²
  4. 4SSW (within groups) = ΣΣ(xᵢⱼ - x̄ᵢ)²
  5. 5F = MSB/MSW

Answer

F = 22.47, p < 0.001. Teaching methods significantly affect scores.

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