Intro to Business Statistics: Descriptive Statistics - Measures of Central, Tendency Mean Median Mode Weighted Mean Trimmed Mean

What This Is

Measures of central tendency are statistical tools used to describe the middle value of a dataset. A retail chain wants to know if average daily sales exceed $10,000 to determine if they should increase inventory. They collect sales data for a month and calculate the mean, median, mode, weighted mean, and trimmed mean to understand their sales pattern.

Key Formulas & Symbols

• Mean (x?): x? = (?x) / n, where x = individual data point, n = sample size.
• Median (M): M = (n + 1)th data point when n is odd, or average of n/2 and (n/2 + 1)th data points when n is even.
• Mode: The most frequently occurring data point.
• Weighted Mean (x?w): x?w = (?wx) / (?w), where w = weights, x = individual data point.
• Trimmed Mean (x?t): x?t = (?(x - c)) / (n - 2c), where c = proportion of data points to trim (e.g., 5%).
• Standard Deviation (?):-= ?[(?(x - ?)^2) / (n - 1)], where-= population mean.
• Variance (?^2): ?^2 = (?(x - ?)^2) / (n - 1).

Step-by-Step Procedure

1. State hypotheses: Formulate null (H?) and alternative (H?) hypotheses.
2. Choose test: Select the appropriate test (e.g., t-test, ANOVA) based on the research question.
3. Compute test statistic: Calculate the test statistic (e.g., t-statistic, F-statistic).
4. Find p-value or critical value: Determine the p-value or critical value using a statistical table or calculator.
5. Compare to ?: Compare the p-value or critical value to the significance level (? = 0.05).
6. Conclude: Make a decision based on the comparison (e.g., reject H?, fail to reject H?).

Common Mistakes

• Mistake: Using Z when-is unknown.
• Correction: Use t-statistic when-is unknown, and degrees of freedom (df) = n - 1.
• Mistake: Misinterpreting p-value as probability H? is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if H? is true.
• Mistake: Failing to check assumptions (e.g., normality, equal variances).
• Correction: Check assumptions before selecting a test and interpreting results.

Quick Practice Problems

1. A company wants to know if their average profit exceeds $100,000. They collect data for a year and calculate the mean profit. If the sample mean is $120,000, the population standard deviation is $50,000, and the sample size is 50, what is the confidence interval?

Answer: $110,000 to $130,000. This is calculated using the formula x? ± (Z * (? / ?n)), where Z = 1.96 for 95% confidence.

1. A marketing firm wants to know if the average age of their customers is 35 years old. They collect data and calculate the mean age. If the sample mean is 40 years old, the population standard deviation is 10 years old, and the sample size is 30, what is the p-value?

Answer: 0.001. This is calculated using the t-test, with df = 29 and t = (40 - 35) / (10 / ?30).

1. A quality control team wants to know if the average weight of their products is 10 pounds. They collect data and calculate the mean weight. If the sample mean is 12 pounds, the population standard deviation is 2 pounds, and the sample size is 20, what is the trimmed mean?

Answer: 11.5 pounds. This is calculated using the formula x?t = (?(x - c)) / (n - 2c), where c = 0.05.

Last-Minute Cram Sheet

1. Mean (x?): x? = (?x) / n.
2. Median (M): M = (n + 1)th data point when n is odd.
3. Mode: The most frequently occurring data point.
4. Weighted Mean (x?w): x?w = (?wx) / (?w).
5. Trimmed Mean (x?t): x?t = (?(x - c)) / (n - 2c).
6. Standard Deviation (?):-= ?[(?(x - ?)^2) / (n - 1)].
7. Variance (?^2): ?^2 = (?(x - ?)^2) / (n - 1).
8. t-statistic: t = (x? - ?) / (? / ?n), where df = n - 1.
9. p-value: The probability of observing the data (or more extreme) if H? is true.
10. ?: The significance level (default = 0.05).
11. p-value is NOT the probability that H? is true – it’s the probability of observing the data (or more extreme) if H? is true.
12. Use t-statistic when-is unknown, and df = n - 1.