Business Statistics: Central Tendency, Dispersion, Correlation and Regression (Basics)

Meaning, scope and importance of business statistics

Business statistics means applying statistical tools to collect, summarize, analyze and interpret business data for decision making.

Scope (exam points):

• collection and classification of data
• presentation (tables, diagrams, graphs)
• averages (central tendency)
• dispersion (variability)
• correlation and regression (relationship between variables)

Importance:

• supports forecasting and planning
• compares performance (departments/years/products)
• reduces uncertainty in decisions

Data, variables and types of data (qualitative/quantitative)

• Data: facts/figures collected for analysis.
• Variable: characteristic that can take different values (sales, price, age).

Types:

• Qualitative (attributes): quality/labels (brand, gender).
• Quantitative (numerical): values (sales, income).

Quantitative further:

• Discrete: counts (number of customers).
• Continuous: measurable (weight, time).

Frequency distribution, class intervals and cumulative frequency

A frequency distribution groups data into classes and shows frequencies.

Key terms:

• Class interval: range like 10–20.
• Class width: size of class (here 10).
• Mid-value (class mark): (lower+upper)/2.
• Cumulative frequency (CF): running total of frequencies.

CF helps find median and quartiles for grouped data.

Measures of central tendency (mean, median, mode)

Central tendency describes a typical/central value.

Arithmetic mean

Mean = (Sum of observations)/n

Merits: simple, uses all values. Demerit: affected by extreme values.

Median

Median is the middle value when data are arranged. Merit: not affected much by extremes.

Mode

Mode is the most frequent value. Useful for “most popular size/brand” type decisions.

Mean calculations (simple, weighted, grouped)

Simple mean

Mean = Σx / n

Weighted mean

Weighted mean = Σ(wx) / Σw Used when items have different importance (quantities, credits).

Mean for grouped data

Mean = Σ(f×m) / Σf where m is class mid-value.

Median and mode for grouped data (steps)

Median (grouped)

Steps:

1. Find total N = Σf.
2. Find N/2.
3. Locate median class (where cumulative frequency just exceeds N/2).
4. Use formula: Median = L + [(N/2 − CF_prev)/f] × h Where:

• L = lower boundary of median class
• CF_prev = cumulative frequency before median class
• f = frequency of median class
• h = class width

Mode (grouped)

Mode = L + [(f1 − f0) / (2f1 − f0 − f2)] × h Where:

• f1 = frequency of modal class
• f0 = frequency of class before it
• f2 = frequency of class after it

Dispersion: range, quartile deviation, variance and standard deviation

Dispersion shows how spread out data are.

Range

Range = Max − Min Simple but depends only on extremes.

Quartile deviation (QD)

QD = (Q3 − Q1)/2 Less affected by extremes.

Variance and standard deviation

For ungrouped data:

• Variance (σ^2) = Σ(x − mean)^2 / n
• Standard deviation (σ) = √(variance)

For grouped data, use f weights: Variance = Σf(x − mean)^2 / Σf

Interpretation: higher SD means more variability/risk.

Coefficient of variation (CV) and comparison

CV compares variability across series with different means: CV = (SD/Mean) × 100

Lower CV → more consistency/stability.

Correlation: meaning, types and coefficient (Pearson basics)

Correlation measures direction and degree of relationship between two variables.

• Positive: both move same direction.
• Negative: move opposite.
• Zero: no linear relationship.

Pearson correlation coefficient r lies between −1 and +1.

• r = +1 perfect positive
• r = −1 perfect negative
• r = 0 no linear correlation

(At basics, focus on interpretation rather than heavy computation.)

Regression: meaning, lines and simple interpretation

Regression estimates one variable based on another.

• Regression of Y on X predicts Y for given X.
• Regression of X on Y predicts X for given Y.

Regression lines are best-fit lines.

Business use: predict sales from advertising, demand from price, etc.

Worked examples (exam-style)

Example 1: Mean

Data: 10, 12, 14, 14, 20 Mean = (10+12+14+14+20)/5 = 70/5 = 14

Example 2: Median and mode

Ordered data: 10, 12, 14, 14, 20 Median = 14 (middle) Mode = 14 (most frequent)

Example 3: Standard deviation (small)

Data: 2, 4, 6 Mean = 4 Deviations: −2, 0, +2 Squares: 4, 0, 4 ⇒ sum = 8 Variance = 8/3 = 2.667 SD = √2.667 ≈ 1.633

Common mistakes and exam tips

• Not arranging data for median.
• Using wrong formula for grouped vs ungrouped data.
• Forgetting to use frequencies in grouped mean/variance.
• Confusing correlation with causation.

Tip: In answers, write formula first, then substitute.

Key terms explained

• Mean/Median/Mode — central values.
• Variance/SD — measure of spread.
• CV — relative variability.
• Correlation — degree of association.
• Regression — prediction relationship.

Quick recap (1-minute revision)

• Mean = Σx/n; weighted mean = Σ(wx)/Σw.
• Median is middle; mode is most frequent.
• SD measures spread; higher SD = higher risk.
• CV compares variability; lower CV = more consistency.
• Correlation shows relationship direction/strength; regression predicts.