Business Statistics CBCBA6T11

Syllabus topics (RTMNU)

• Correlation- Meaning and types of correlation
• Scatter diagram method
• Karl Pearson’s coefficient of correlation
• Spearman’s Rank correlation

Meaning of Correlation

Correlation is a statistical tool that measures the degree and direction of relationship between two variables. When a change in one variable is accompanied by a change in the other, the two are said to be correlated. Examples: price and demand, advertising expenditure and sales, income and expenditure.

The coefficient of correlation (r) lies between −1 and +1. r = +1: perfect positive; r = −1: perfect negative; r = 0: no linear correlation.

Types of Correlation

1. Positive and negative: positive when both variables move in the same direction (income and consumption); negative when they move in opposite directions (price and demand).
2. Linear and non-linear (curvilinear): linear when the ratio of change is constant (points lie on a straight line); non-linear when the ratio changes.
3. Simple, partial and multiple: simple – two variables; multiple – three or more studied together; partial – two variables studied keeping others constant.

Scatter Diagram Method

A scatter diagram is a graph in which pairs of values (X, Y) are plotted as dots. The pattern shows the type of correlation: dots rising from left to right – positive; falling – negative; all on a straight line – perfect; scattered with no pattern – no correlation; close to a line – high degree.

Merits: simple, easy to understand, not affected by extreme values, first step in study. Demerits: does not give an exact numerical value; only a rough idea.

Karl Pearson's Coefficient of Correlation

Also called the product moment correlation coefficient. Formula (actual mean method):

Properties: lies between −1 and +1; independent of change of origin and scale; it is the geometric mean of the two regression coefficients (r = ±√(b_xy × b_yx)); it is a pure number (no unit).

Illustration 1: Find Karl Pearson's coefficient of correlation between advertising expenditure X (₹ thousand) and sales Y (₹ lakh): X: 10, 12, 14, 16, 18, 20; Y: 15, 18, 20, 25, 28, 32.

X̄ = 90/6 = 15; Ȳ = 138/6 = 23.

r = 120 ÷ √(70 × 208) = 120 ÷ √14,560 = 120 ÷ 120.67 = 0.994. There is a very high degree of positive correlation between advertising and sales.

Spearman's Rank Correlation

Developed by Charles Edward Spearman (1904). Used when data are qualitative (beauty, honesty, ability) or given as ranks.

Illustration 2: Two judges gave marks to 6 students. Find rank correlation. Judge A: 85, 60, 73, 40, 90, 50; Judge B: 75, 65, 80, 45, 70, 55 (highest marks = rank 1).

R = 1 − (6 × 8) ÷ [6 × (36 − 1)] = 1 − 48/210 = 1 − 0.229 = 0.771. The judges agree to a high degree.

Merits: simple, useful for qualitative data. Demerits: not suitable for large data or grouped frequency data; less accurate than Pearson's method.

Likely exam questions

1. (Long, 8/16 marks) What is correlation? Explain its types and the scatter diagram method.
2. (Long, 8/16 marks) Calculate Karl Pearson's coefficient of correlation: X: 2, 4, 6, 8, 10; Y: 4, 7, 9, 12, 13.
3. (Short, 4 marks) Give any four properties of Karl Pearson's coefficient of correlation.
4. (Short, 4 marks) Calculate rank correlation if ΣD² = 30 and N = 10.
5. (Short, 4 marks) Distinguish between positive and negative correlation.

Syllabus topics (RTMNU)

• Meaning, Nature, scope and importance of regression analysis
• Regression equations - X on Y & Y on X
• Estimation of regression coefficients using different methods

Meaning and Nature of Regression Analysis

Regression analysis is a statistical method of estimating (predicting) the unknown value of one variable from the known value of another variable, on the basis of the average relationship between them. The word "regression" (going back) was first used by Sir Francis Galton in his study of heights of fathers and sons.

• The variable that is predicted is the dependent variable; the variable used to predict is the independent variable.
• A regression line is the line of best fit, drawn by the method of least squares.
• With two variables there are two regression lines: Y on X (to estimate Y from X) and X on Y (to estimate X from Y). They meet at the point of means (X̄, Ȳ).

Scope and Importance

1. Forecasting in business: estimating sales from advertising, demand from price, cost from output.
2. Economics: studying cause-and-effect relationships (consumption and income).
3. Planning and policy: government estimates tax revenue, production targets.
4. Measuring error of estimate through the standard error of estimate.
5. Finds correlation: r = ±√(b_xy × b_yx).

Regression Equations: Y on X and X on Y

Properties of regression coefficients: (1) both have the same sign as r; (2) r is the geometric mean of the two; (3) both cannot be more than 1 together; (4) they are independent of change of origin but not of scale.

Estimation of Regression Coefficients by Different Methods

1. Actual mean method: b_yx = Σxy/Σx² (when means are whole numbers).
2. Assumed mean method: b_yx = [NΣdxdy − ΣdxΣdy] ÷ [NΣdx² − (Σdx)²] (when means are in fractions).
3. From r and standard deviations: b_yx = r σ_y/σ_x; b_xy = r σ_x/σ_y.
4. Normal equations (least squares): for Y = a + bX: ΣY = Na + bΣX and ΣXY = aΣX + bΣX².

Illustration 1 (actual mean method): Using the advertising (X) and sales (Y) data of Unit I – X: 10, 12, 14, 16, 18, 20; Y: 15, 18, 20, 25, 28, 32 – where X̄ = 15, Ȳ = 23, Σx² = 70, Σy² = 208, Σxy = 120. Find both regression equations and estimate sales when advertising is 22.

b_yx = 120/70 = 1.714; b_xy = 120/208 = 0.577.

Y on X: Y − 23 = 1.714 (X − 15) → Y = 1.714X − 25.71 + 23 → Y = 1.714X − 2.71.

X on Y: X − 15 = 0.577 (Y − 23) → X = 0.577Y − 13.27 + 15 → X = 0.577Y + 1.73.

Estimate: when X = 22, Y = 1.714 × 22 − 2.71 = 37.71 − 2.71 = 35.0 (₹ lakh).

Check: r = √(1.714 × 0.577) = √0.989 = 0.994 – same as Unit I. ✔

Illustration 2 (from r and σ): X̄ = 40, Ȳ = 50, σ_x = 10, σ_y = 16, r = 0.6. Find the regression equations and the most likely value of Y when X = 50.

b_yx = 0.6 × 16/10 = 0.96; b_xy = 0.6 × 10/16 = 0.375.

Y on X: Y − 50 = 0.96 (X − 40) → Y = 0.96X + 11.6. X on Y: X − 40 = 0.375 (Y − 50) → X = 0.375Y + 21.25.

When X = 50: Y = 0.96 × 50 + 11.6 = 48 + 11.6 = 59.6.

Likely exam questions

1. (Long, 8/16 marks) Explain the meaning and importance of regression analysis. Distinguish between correlation and regression.
2. (Long, 8/16 marks) Given X̄ = 25, Ȳ = 40, σx = 5, σy = 8 and r = 0.75, find both regression equations and estimate Y when X = 30.
3. (Short, 4 marks) State the properties of regression coefficients.
4. (Short, 4 marks) If byx = 0.8 and bxy = 0.45, find r.
5. (Short, 4 marks) Why are there two regression lines?

Syllabus topics (RTMNU)

• Meaning and importance of time series in business forecasting
• Components of time series- Trend, Seasonal variation, Cyclical variation, Irregular variation
• Methods of measuring trend- Moving average method, least square method
• Applications of time series in business planning and forecasting

Meaning and Importance of Time Series

A time series is a set of observations of a variable arranged in order of time (years, quarters, months, days). Examples: annual sales of Maruti Suzuki, monthly GST collection, daily Sensex closing values.

Importance in business forecasting:

1. Helps understand past behaviour of a variable.
2. Helps forecast future sales, demand, prices and production.
3. Helps compare actual performance with expected (planning and control).
4. Separates the effect of each component (trend, seasons, cycles), so that decisions are better (e.g., stock planning before Diwali).
5. Useful to economists, government (budget planning) and investors.

Components of Time Series

1. Secular trend (T): the general long-term tendency to increase, decrease or remain steady (e.g., rising population, growing online sales).
2. Seasonal variation (S): regular, short-term changes that repeat within a year due to weather, festivals and customs (e.g., sale of umbrellas in monsoon, sweets and clothes at Diwali).
3. Cyclical variation (C): up-and-down movements lasting more than a year, due to business cycles – prosperity, recession, depression and recovery.
4. Irregular (random) variation (I): unpredictable changes due to floods, strikes, wars, pandemics (e.g., COVID-19 lockdown).

Models: Additive model Y = T + S + C + I; Multiplicative model Y = T × S × C × I.

Methods of Measuring Trend

Main methods: (1) freehand/graphic method, (2) semi-average method, (3) moving average method, (4) method of least squares. Your syllabus stresses the last two.

Moving Average Method

In this method, the average of a fixed number of consecutive years (the period, e.g., 3 or 5 years) is calculated, moving one year at a time. Each average is placed against the middle year and is the trend value for that year. For an even period (4 years), the moving averages are centred by taking a 2-item moving average again.

Merits: simple; removes short-term fluctuations; flexible. Demerits: trend values are lost at the beginning and end; no equation, so cannot forecast; choice of period is subjective.

Illustration 1: Calculate the 3-yearly moving averages for the sales of a firm.

(e.g., 70 + 74 + 80 = 224; 224 ÷ 3 = 74.67, placed against 2020.)

Method of Least Squares

The trend line Y_c = a + bx is fitted so that the sum of squares of deviations of actual values from trend values is minimum, and the sum of deviations is zero. It is the best-fitting line.

Illustration 2: Fit a straight-line trend to the same sales data by the method of least squares, find trend values and estimate sales for 2027.

a = 573/7 = 81.86; b = 113/28 = 4.04. Trend equation: Y_c = 81.86 + 4.04x (origin: 2022; x unit: 1 year; Y in ₹ lakh).

Estimate for 2027 (x = 5): Y = 81.86 + 4.04 × 5 = ₹102.04 lakh (using exact b = 4.0357).

Merits: mathematical and objective; gives trend for all years; allows forecasting. Demerits: calculations are longer; adding new data changes the whole line; assumes a straight-line trend.

Applications of Time Series in Business Planning and Forecasting

• Sales forecasting for production and inventory planning.
• Seasonal planning – staffing, raw material buying, advertising timing.
• Financial planning – cash budgets, capital budgeting based on expected revenue.
• Economic policy – RBI and government study trends in prices, output, employment.
• Stock market – technical analysts use moving averages of share prices.

Likely exam questions

1. (Long, 8/16 marks) What is a time series? Explain its components with examples.
2. (Long, 8/16 marks) Fit a straight-line trend by the method of least squares and estimate the value for 2026: 2019–2023 production (tonnes): 40, 44, 46, 50, 55.
3. (Short, 4 marks) Give merits and demerits of the moving average method.
4. (Short, 4 marks) What is seasonal variation?
5. (Short, 4 marks) Calculate 3-yearly moving averages: 12, 15, 14, 18, 20, 19.

Syllabus topics (RTMNU)

• Meaning and importance of index numbers
• Types of index numbers
• Methods of constructing index numbers-Simple price index, Weighted price index
• Consumer Price Index (Cost of Living Index)
• Uses and limitations of index numbers in business

Meaning and Importance of Index Numbers

An index number is a statistical device that measures the relative (percentage) change in a variable or a group of related variables over time or between places, compared to a base. The base year index is taken as 100. Example: if the price index for 2025 is 125 with 2015 as base, prices have risen by 25%.

Index numbers are called "economic barometers" because they show the condition of the economy.

Importance: measure changes in price level and inflation; help government policy (RBI uses CPI inflation in monetary policy); used to fix dearness allowance (DA) of employees; show trends in production, trade and the share market (Sensex, Nifty are index numbers); help businesses in planning and comparison.

Types of Index Numbers

1. Price index numbers: changes in prices – Wholesale Price Index (WPI), Consumer Price Index (CPI).
2. Quantity index numbers: changes in quantity produced or sold – Index of Industrial Production (IIP).
3. Value index numbers: changes in total value (price × quantity).
4. Special purpose indices: share price indices (BSE Sensex, NSE Nifty 50), cost of living index.

By method: unweighted (simple) – all items equally important; weighted – items given importance by quantity or value.

Methods of Constructing Index Numbers: Simple Price Index

Weighted Price Index Numbers

Fisher's index is called "ideal" because it uses both base and current quantities, is free from bias, and satisfies the time reversal test and the factor reversal test.

Illustration 1: Calculate the price index for 2025 (base 2020) by (a) simple aggregative method, (b) simple average of price relatives, (c) Laspeyres', (d) Paasche's and (e) Fisher's methods.

(a) Simple aggregative = 53/43 × 100 = 123.26.

(b) Simple average of relatives = 490/4 = 122.50.

(c) Laspeyres' = 258/210 × 100 = 122.86.

(d) Paasche's = 272/222 × 100 = 122.52.

(e) Fisher's = √(122.86 × 122.52) = 122.69. Prices rose by about 22.7%.

Consumer Price Index (Cost of Living Index)

The Consumer Price Index (CPI) measures the average change in the retail prices of a fixed basket of goods and services consumed by a particular group of people (e.g., industrial workers, agricultural labourers, urban or rural households). It shows the change in their cost of living. In India, CPI (Rural, Urban, Combined) is published by the National Statistics Office (MoSPI), and CPI-IW by the Labour Bureau. (Base years are revised from time to time – check the latest base year.)

Steps: (1) decide the group of people; (2) family budget survey to find items consumed and their weights; (3) collect retail prices; (4) compute group indices; (5) combine using weights.

Illustration 2: Find the cost of living index from the following group indices and weights.

CPI = 16,790 ÷ 100 = 167.9. The cost of living has risen by 67.9% over the base year. If a worker earned ₹15,000 in the base year, he needs 15,000 × 167.9/100 = ₹25,185 now to keep the same standard of living.

Uses of CPI: fixing DA and wages; measuring purchasing power of money (= 100/CPI); RBI's inflation target is set in terms of CPI inflation; deflating income to find real income.

Uses and Limitations of Index Numbers in Business

Uses: measure inflation; help in wage negotiation; help in pricing and budgeting; compare business performance over time; deflate sales figures to real terms; guide investment decisions (stock indices).

Limitations:

1. Based on samples, so not fully accurate.
2. Quality changes and new products are hard to include.
3. Choice of base year and weights may cause bias.
4. An index made for one purpose (or group) may not suit another.
5. Shows only relative change, not absolute change.

Likely exam questions

1. (Long, 8/16 marks) What are index numbers? Explain their types, uses and limitations.
2. (Long, 8/16 marks) Compute Laspeyres', Paasche's and Fisher's price index: Item A p0 4, q0 10, p1 6, q1 12; Item B p0 5, q0 8, p1 6, q1 10; Item C p0 10, q0 5, p1 12, q1 4.
3. (Short, 4 marks) Why is Fisher's index called ideal?
4. (Short, 4 marks) What is the Consumer Price Index?
5. (Short, 4 marks) Calculate the cost of living index if ΣWI = 13,450 and ΣW = 100.