Business Statistics I: Descriptive Statistics, Probability, and Distributions

❦

The NIE Business Statistics syllabus (introduced in 1995 and revised for 2017) develops the use of statistics in business decisions, including descriptive statistics, probability, sampling, regression, and time series (NIE, n.d.-h). Texts such as Levine et al. (2017), Anderson et al. (2020), and Keller (2018) provide further examples.

Why Statistics?

Statistics supports business decisions by collecting, summarizing, and analyzing data under uncertainty. Descriptive statistics summarize data; inferential statistics draw conclusions about populations from samples. Be aware of the misuse of statistics: misleading graphs, biased samples, and confusing correlation with causation.

Data and Sampling

Types of data: qualitative (nominal, ordinal) and quantitative (discrete, continuous); scales of measurement; primary and secondary data.
Sampling: probability sampling (simple random, systematic, stratified, cluster) and non-probability sampling (convenience, judgment, quota). Sources of error: sampling error, non-response, and measurement error.
Presenting data: tables, bar charts, pie charts, histograms, frequency polygons, ogives, scatter plots; good graphs have a title, labels, a suitable scale, and no distortion.

Measures of Location and Spread

Measure	Formula or description
Mean	x̄ = Σx ÷ n (for frequency data, Σfx ÷ Σf)
Median	The middle value of ordered data
Mode	The most frequent value
Range; interquartile range	Maximum − minimum; Q₃ − Q₁
Variance	s² = Σ(x − x̄)² ÷ (n − 1) for a sample (σ² = Σ(x − μ)² ÷ N for a population)
Standard deviation	s = √variance
Coefficient of variation	(s ÷ x̄) × 100, for comparing variability
Skewness	The direction of the tail; mean > median for positive skew

Worked example. Daily sales (in thousands): 4, 6, 8, 10, 12. Mean = 8. Deviations: −4, −2, 0, 2, 4; squares: 16, 4, 0, 4, 16 (sum 40). Sample variance = 40 ÷ 4 = 10; standard deviation ≈ 3.16; CV = 3.16 ÷ 8 × 100 ≈ 39.5 percent.

Index Numbers

Index numbers measure changes in prices or quantities over time against a base period (base = 100).

Simple price relative = (p₁ ÷ p₀) × 100.
Laspeyres price index = Σp₁q₀ ÷ Σp₀q₀ × 100 (base-period quantities).
Paasche price index = Σp₁q₁ ÷ Σp₀q₁ × 100 (current-period quantities).
The Consumer Price Index is a Laspeyres-type index. Use indices to deflate nominal values: real value = nominal value ÷ price index × 100.

Time Series

Components: trend (long-term direction), seasonal variation, cyclical variation, and irregular (random) variation. Moving averages smooth data and estimate the trend; seasonal indices are found by comparing actual values with the trend (additive or multiplicative models); forecasting uses the trend and seasonal factors.

Probability

Classical (equally likely outcomes), relative frequency, and subjective probability. Addition rule: P(A or B) = P(A) + P(B) − P(A and B). Multiplication rule for independent events: P(A and B) = P(A) × P(B). Conditional probability: P(A|B) = P(A and B) ÷ P(B). Bayes' theorem updates probabilities with new information.
Business application: the chance that a customer defaults, a machine fails, or a campaign succeeds.

Probability Distributions

Distribution	Use	Key formulas
Binomial	The number of successes in n independent trials, each with probability p	P(X = r) = ⁿCᵣ pʳ (1 − p)ⁿ⁻ʳ; mean np; variance np(1 − p)
Poisson	Events in a fixed interval (for example calls per hour)	P(X = k) = e^(−λ) λᵏ ÷ k!; mean = variance = λ
Normal	Continuous, symmetric, bell-shaped data	z = (x − μ) ÷ σ; about 68, 95, and 99.7 percent of values lie within 1, 2, and 3 standard deviations

Example (binomial): with n = 5 and p = 0.2, P(X = 2) = 10 × 0.04 × 0.512 = 0.2048. Use standard normal tables to find probabilities and percentiles; expected value = Σ x P(x).

Common Mistakes

Using the population formula for sample variance.
Misreading the base year in index numbers.
Applying the normal distribution to clearly skewed data without checking.

CHAPTER 17