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Calculating Statistics

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Calculating Statistics

Complete Guide to Calculating Statistics

Master statistical calculations with comprehensive step-by-step tutorials, formulas, and practical examples for IB, AP, GCSE, and university-level mathematics. Learn to calculate measures of central tendency, variability, correlation, and hypothesis testing.

1 Calculating Measures of Central Tendency

μ Mean (Average) Calculation

Step-by-Step Process:

  1. Add all values in your dataset
  2. Count the number of values (n)
  3. Divide the sum by the count
Formula:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n} \]

📝 Worked Example:

Data: 4, 7, 9, 10, 15

Step 1: Sum = 4 + 7 + 9 + 10 + 15 = 45

Step 2: Count (n) = 5

Step 3: Mean = 45 ÷ 5 = 9

M Median Calculation

Step-by-Step Process:

  1. Arrange data in ascending order
  2. Find the middle position(s)
  3. Identify median value based on n
Formulas:
If n is odd: \[ \text{Position} = \frac{n+1}{2} \]
If n is even: \[ \text{Median} = \frac{x_{n/2} + x_{(n/2)+1}}{2} \]

📝 Examples:

Odd n: 3, 7, 9, 12, 15 → Median = 9 (middle value)

Even n: 3, 7, 9, 12 → Median = (7+9)/2 = 8

Mo Mode Calculation

Step-by-Step Process:

  1. Count frequency of each value
  2. Identify the most frequent value(s)
  3. Note: Can have no mode, one mode, or multiple modes

📝 Examples:

Unimodal: 2, 3, 3, 4, 5 → Mode = 3

Bimodal: 1, 2, 2, 3, 3, 4 → Modes = 2, 3

No mode: 1, 2, 3, 4, 5 → No repeating values

R Range Calculation

Step-by-Step Process:

  1. Identify the highest value (maximum)
  2. Identify the lowest value (minimum)
  3. Subtract: Range = Maximum - Minimum
Formula:
\[ \text{Range} = x_{\max} - x_{\min} \]

📝 Worked Example:

Data: 15, 22, 8, 35, 12, 18

Maximum: 35

Minimum: 8

Range: 35 - 8 = 27

2 Calculating Measures of Variability

σ² Variance Calculation - Step by Step

📋 Step-by-Step Process:

  1. Calculate the mean (x̄)
  2. Find deviations from mean for each value: (x - x̄)
  3. Square each deviation: (x - x̄)²
  4. Sum all squared deviations: Σ(x - x̄)²
  5. Divide by: n-1 (sample) or n (population)

📐 Formulas:

Sample Variance: \[ s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} \]
Population Variance: \[ \sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N} \]

🔍 Detailed Worked Example:

Data: 4, 7, 9, 10, 15

Step 1: Mean
x̄ = (4+7+9+10+15)/5 = 9
Step 2: Deviations
4-9=-5, 7-9=-2, 9-9=0
10-9=1, 15-9=6
Step 3: Squared
25, 4, 0, 1, 36
Step 4: Sum
Σ(x-x̄)² = 25+4+0+1+36 = 66
Step 5: Final Answer
Sample Variance: s² = 66/(5-1) = 66/4 = 16.5

σ Standard Deviation Calculation

✨ Simple Process:

  1. Calculate variance first (follow steps above)
  2. Take the square root of the variance
Key Point: Standard deviation is in the same units as the original data, while variance is in squared units.

📐 Formulas:

Sample Standard Deviation: \[ s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} \]
Population Standard Deviation: \[ \sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}} \]

📝 Continuing Previous Example:

From our variance calculation: s² = 16.5

Standard Deviation: s = √16.5 ≈ 4.06

3 Calculating Correlation Coefficient

r Pearson Correlation Coefficient

📋 Step-by-Step Process:

  1. Calculate means of both variables (x̄, ȳ)
  2. Find deviations from means
  3. Calculate products of deviations
  4. Calculate standard deviations of both variables
  5. Apply correlation formula

📐 Formula:

\[ r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} \]

Alternative: r = Cov(x,y) / (sx × sy)

🔍 Worked Example:

Data: Hours studied (x): 2, 4, 6, 8 | Test scores (y): 65, 70, 85, 90

xyx - x̄y - ȳ(x-x̄)(y-ȳ)(x-x̄)²(y-ȳ)²
265-3-12.537.59156.25
470-1-7.57.5156.25
68517.57.5156.25
890312.537.59156.25
Sums:009020425

Calculation:

x̄ = 5, ȳ = 77.5

r = 90 / √(20 × 425) = 90 / √8500 = 90 / 92.2 ≈ 0.976

Interpretation: Very strong positive correlation

📊 Correlation Interpretation Guide:

Strong Negative: -1.0 to -0.7
Moderate Negative: -0.7 to -0.3
Weak/No Correlation: -0.3 to 0.3
Moderate Positive: 0.3 to 0.7
Strong Positive: 0.7 to 1.0

4 Calculating Test Statistics

H General Hypothesis Testing Steps

  1. State Hypotheses:
    • H₀ (null hypothesis)
    • H₁ (alternative hypothesis)
  2. Choose significance level (α) (typically 0.05)
  3. Calculate test statistic using appropriate formula
  4. Find p-value or critical value
  5. Make decision: Reject or fail to reject H₀
Decision Rule:
If p-value ≤ α: Reject H₀
If p-value > α: Fail to reject H₀

Z Z-Test Calculation

When to Use:

  • Population standard deviation is known
  • Large sample size (n ≥ 30)
  • Population is normally distributed
One-Sample Z-Test Formula:
\[ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \]

📝 Example:

Sample mean = 52, μ = 50, σ = 8, n = 64

z = (52 - 50)/(8/√64) = 2/(8/8) = 2/1 = 2.0

t T-Test Calculation

When to Use:

  • Population standard deviation is unknown
  • Small sample size (n < 30)
  • Use sample standard deviation
One-Sample T-Test Formula:
\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]

Degrees of freedom: df = n - 1

📝 Example:

Sample mean = 18.5, μ = 20, s = 3.2, n = 16

t = (18.5 - 20)/(3.2/√16) = -1.5/(3.2/4) = -1.5/0.8 = -1.875

df = 16 - 1 = 15

t₂ Two-Sample T-Test

Formula:
\[ t = \frac{\bar{x_1} - \bar{x_2}}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]
Pooled Standard Deviation:
\[ s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}} \]

Degrees of freedom: df = n₁ + n₂ - 2

🖩 Calculator Tips & Common Mistakes

📱 Calculator Tips

  • STAT mode: Use for entering data lists
  • 1-Var Stats: Calculates mean, std dev, variance
  • 2-Var Stats: For correlation analysis
  • Check n vs n-1: Sample vs population formulas
  • Store intermediate results to avoid rounding errors

⚠️ Common Mistakes

  • Confusing sample (n-1) vs population (n) formulas
  • Forgetting to square deviations for variance
  • Not arranging data for median calculation
  • Using wrong test (z vs t) based on conditions
  • Misinterpreting correlation vs causation

💡 Study Tips

  • Practice by hand first to understand concepts
  • Verify calculator results with simple examples
  • Understand when to use each statistical measure
  • Check assumptions before applying tests
  • Interpret results in context of the problem

🔍 Quick Reference

Mean: \( \bar{x} = \frac{\sum x}{n} \)

Variance: \( s^2 = \frac{\sum(x-\bar{x})^2}{n-1} \)

Std Dev: \( s = \sqrt{s^2} \)

Z-score: \( z = \frac{x-\mu}{\sigma} \)

Correlation: \( r = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sqrt{\sum(x-\bar{x})^2\sum(y-\bar{y})^2}} \)

About the Author

Adam Kumar

Co-Founder @RevisionTown
Mathematics Expert specializing in various curricula including IB, AP, GCSE, IGCSE, and more. Dedicated to creating comprehensive statistical education resources and step-by-step calculation guides for students worldwide.

RevisionTown provides comprehensive study materials, interactive calculators, and step-by-step guides for mathematics and statistics across multiple international curricula.

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