Intro to Business Statistics: Descriptive Statistics Measures of Dispersion Range Variance Population vs Sample Standard Deviation Coefficient of Variation

What This Is

Measures of dispersion, also known as variability, are essential in business decisions as they help quantify the spread of data. A retail chain wants to know if average daily sales exceed $10,000 to determine if they need to adjust their inventory levels. By calculating measures of dispersion, they can understand the consistency of their sales and make informed decisions.

Key Formulas & Symbols

• Range (R) = Maximum value - Minimum value where Maximum value and Minimum value are the largest and smallest values in the dataset.
• Variance (σ²) = Σ(xi - μ)² / (n - 1) where xi = individual data points, μ = population mean, n = sample size.
• Sample Variance (s²) = Σ(xi - x̄)² / (n - 1) where xi = individual data points, x̄ = sample mean, n = sample size.
• Standard Deviation (σ) = √(σ²) where σ² = population variance.
• Sample Standard Deviation (s) = √(s²) where s² = sample variance.
• Coefficient of Variation (CV) = (σ / μ) × 100 where σ = population standard deviation, μ = population mean.
• Z = (x̄ - μ) / (σ/√n) where x̄ = sample mean, μ = population mean, σ = population standard deviation, n = sample size.
• t = (x̄ - μ) / (s/√n) where x̄ = sample mean, μ = population mean, s = sample standard deviation, n = sample size.

Step-by-Step Procedure

1. State hypotheses: Clearly define the null and alternative hypotheses (e.g., H₀: μ = 10,000 vs. H₁: μ ≠ 10,000).
2. Choose test: Select the appropriate test (e.g., Z-test for known population standard deviation, t-test for unknown population standard deviation).
3. Compute test statistic: Calculate the test statistic (e.g., Z or t) using the given formula.
4. Find p-value or critical value: Determine the p-value or critical value using a standard normal distribution (Z-table) or t-distribution table.
5. Compare to α: Compare the p-value or critical value to the significance level (α = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H₀ if p-value < α).

Common Mistakes

• Mistake: Using Z when σ is unknown.
• Correction: Use t-test instead, as it accounts for sample standard deviation.
• Mistake: Misinterpreting p-value as probability H₀ is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H₀ is true.
• Mistake: Failing to check assumptions (e.g., normality, equal variances).
• Correction: Verify assumptions before selecting a test.

Quick Practice Problems

1. A company wants to know if their average daily sales exceed $10,000. The sample mean is $12,000, sample standard deviation is $2,000, and sample size is 50. Calculate the confidence interval.

Answer: ($10,000, $14,000) Explanation: Using the t-distribution, we calculate the confidence interval as x̄ ± (t * (s / √n)).

1. A quality control team wants to determine if the average weight of a product is 5 kg. The sample mean is 4.8 kg, sample standard deviation is 0.2 kg, and sample size is 36. What is the p-value?

Answer: 0.02 Explanation: We calculate the t-statistic and use a t-distribution table to find the p-value.

1. A marketing team wants to know if the average response rate to an advertisement is 2%. The sample mean is 1.8%, sample standard deviation is 0.5%, and sample size is 100. Calculate the coefficient of variation.

Answer: 25% Explanation: We calculate the CV as (σ / μ) × 100.

Last-Minute Cram Sheet

1. Z-test: Use for known population standard deviation.
2. t-test: Use for unknown population standard deviation.
3. p-value: Probability of observing the data (or more extreme) if H₀ is true.
4. α = 0.05: Default significance level.
5. σ² = Σ(xi - μ)² / (n - 1): Population variance formula.
6. s² = Σ(xi - x̄)² / (n - 1): Sample variance formula.
7. σ = √(σ²): Population standard deviation formula.
8. s = √(s²): Sample standard deviation formula.
9. CV = (σ / μ) × 100: Coefficient of variation formula.
10. ⚠️ p-value is NOT the probability that H₀ is true – it’s the probability of observing the data (or more extreme) if H₀ is true.