Numerical Measures in Business Statistics

Numerical Measures in Business Statistics

Outline and Learning Outcomes

• Outline:

  • What are measures?

  • Measures of Data Center and Location

  • Measures of Data Variation

• Learning Outcomes: Students will be able to:

  • Compute and describe data using the mean, median, mode, and weighted mean.

  • Compute and describe data using range, interquartile range, variance, and standard deviation.

  • Compute and use z-score and the coefficient of variation to describe data.

  • Apply the Empirical Rule and Tchebysheff's theorem.

Measures of Center and Location

• Measure:

  • A quantitative value that describes a particular characteristic of a dataset.

  • Useful for summarizing and interpreting data.

  • Allows complex datasets to be distilled into a single useful number.

Key Terms: Parameter and Statistic

• Parameter:

  • A measure computed from the entire population.

  • This quantitative value is constant if the population does not change.

  • Usually denoted with a Greek character (e.g., $\mu$ for population mean, $\sigma$ for population standard deviation).

• Statistic:

  • A measure computed from a sample of a population.

  • Will vary based on the specific sample taken.

  • Usually denoted with a Roman character (e.g., $\bar{x}$ for sample mean, $s$ for sample standard deviation).

Mean (Average)

• The most commonly used measure of central tendency.

• Helps to describe the center of a dataset.

• =average(range) in Excel.

Population Mean ($\mu$)

• A parameter.

• Formula: $\mu = \frac{\sum<em>{i=1}^{N} x</em>i}{N}$

  • $\mu$: Population mean (pronounced "mu")

  • $N$: Population size

  • $x_i$: $i^{th}$ individual value of the variable

• Example #1: Sales prices: $\$9,000, \ \$10,000, \ \$21,000$. For $N=3$.

  • $\mu = \frac{\$9,000 + \$10,000 + \$21,000}{3} = \frac{\$40,000}{3} = \$13,333.33$

• Example #2: Sales prices: $\$8,000, \ \$9,000, \ \$10,000, \ \$41,000$. For $N=4$.

  • $\mu = \frac{\$8,000 + \$9,000 + \$10,000 + \$41,000}{4} = \frac{\$68,000}{4} = \$17,000$

Sample Mean ($\bar{x}$)

• A statistic.

• Formula: $\bar{x} = \frac{\sum<em>{i=1}^{n} x</em>i}{n}$

  • $\bar{x}$: Sample mean (pronounced "x-bar")

  • $n$: Sample size

  • $x_i$: $i^{th}$ individual value of the variable

• Example #1: Cars sold daily: $28, 12, 6, 4, 5, 15, 10$. For $n=7$.

  • $\bar{x} = \frac{28+12+6+4+5+15+10}{7} = \frac{80}{7} \approx 11.43$ cars

Median

• Another center measure, less impacted by outliers than the mean.

• Divides a data array into two equal halves.

• =median(range) in Excel.

• To Compute the Median:

  1. Sort the data in ascending order.

  2. Calculate the median's index ($i$).

     • For population data index ($N$): $i = rac{1}{2}N$

     • For sample data index ($n$): $i = rac{1}{2}n$

  3. If index $i$ is not an integer: Round up to the next integer. The median is the value at this rounded index position.

  4. If index $i$ is an integer: The median is the average of the values in index positions $i$ and $i+1$.

• Example #1 (Odd N): Prices: $\$9,000, \ \$10,000, \ \$21,000$. ($N=3$)

  1. Sorted: $\$9,000, \ \$10,000, \ \$21,000$

  2. Index: $i = \frac{1}{2}(3) = 1.5$. Round up to $2$.

  3. Median is the $2^{nd}$ value: $\$10,000$.

• Example #2 (Odd N): Cars sold: $4, 5, 6, 10, 12, 15, 28$. (Sorted, $n=7$)

  1. Index: $i = \frac{1}{2}(7) = 3.5$. Round up to $4$.

  2. Median is the $4^{th}$ value: $10$ cars.

• Example #3 (Even N): Cars sold: $4, 5, 6, 10, 12, 15, 28, 500$. (Sorted, $n=8$)

  1. Index: $i = \frac{1}{2}(8) = 4$. (Integer)

  2. Median is the average of values at positions $4$ and $5$.

  3. Values: $10$ and $12$. Median $= \frac{10+12}{2} = 11$ cars.

Using Mean and Median Together

• Comparing the mean and median can reveal insights about the dataset's distribution, especially regarding extreme values:

  • Mean > Median: The dataset is skewed to the right (positive skew), indicating the presence of high extreme values.

  • Mean < Median: The dataset is skewed to the left (negative skew), indicating the presence of low extreme values.

  • Mean = Median: The dataset is evenly spread and symmetric.

Mode

• Another measure of central location, though not as common as mean or median.

• Counts the most frequent value(s) in the dataset.

• A dataset can have multiple modes (bimodal, multimodal) or no mode.

• Note: The mode need not be in the center and may not always reflect the center of the data set; interpretation requires care.

• Example #1 (Single Mode): Daily car sales: $3, 3, 4, 5, 7, 7, 7, 9, 10, 10, 11, 11, 13, 13, 15$

  • The value $7$ appears 3 times, which is more than any other value. Mode = $7$.

• Example #2 (Multiple Modes): Song plays: $0, 0, 1, 1, 1, 2, 3, 5, 5, 5$

  • The values $1$ and $5$ both appear 3 times. Modes = $1, 5$.

• Center Considerations:

  • Datasets with no value(s) occurring more frequently than others have no mode.

  • The mode may not necessarily be close to the mean or median (e.g., for dataset ${3, 3, 3, 3, 3, 8, 9, 12, 16, 16, 16}$, mode is 3, median is 8, mean is 8.4).

Other Location Measures

Weighted Mean

• Used when some data values are more important or occur more frequently than others.

• Also known as weighted average.

Weighted Mean for a Population ($\mu_w$)

• Formula: $\mu<em>w = \frac{\sum w</em>i x<em>i}{\sum w</em>i}$

  • $w_i$: The weight of the $i^{th}$ data value

  • $x_i$: The $i^{th}$ data value

Weighted Mean for a Sample ($\bar{x}_w$)

• Formula: $\bar{x}<em>w = \frac{\sum w</em>i x<em>i}{\sum w</em>i}$

  • $w_i$: The weight of the $i^{th}$ data value

  • $x_i$: The $i^{th}$ data value

• Example #1 (Income):

  • Worker 1: 50 days, $\$1000$. $w<em>1 = 50, x</em>1 = 1000$

  • Worker 2: 20 days, $\$500$. $w<em>2 = 20, x</em>2 = 500$

  • Worker 3: 30 days, $\$800$. $w<em>3 = 30, x</em>3 = 800$

  • $\bar{x}_w = \frac{(50 \times \$1000) + (20 \times \$500) + (30 \times \$800)}{50+20+30} = \frac{\$50,000 + \$10,000 + \$24,000}{100} = \frac{\$84,000}{100} = \$840$

• Example #2 (Antique Store):

  • Cust 1: 5 items, $\$100$. $w<em>1 = 5, x</em>1 = 100$

  • Cust 2: 3 items, $\$50$. $w<em>2 = 3, x</em>2 = 50$

  • Cust 3: 4 items, $\$80$. $w<em>3 = 4, x</em>3 = 80$

  • $\bar{x}_w = \frac{(5 \times \$100) + (3 \times \$50) + (4 \times \$80)}{5+3+4} = \frac{\$500 + \$150 + \$320}{12} = \frac{\$970}{12} \approx \$80.83$

Percentiles

• Bin data into equally spaced percentage bins.

• Can answer questions like: