व्यावसायिक गणित व सांख्यिकी Business Mathematics and Statistics

थोडक्यात सारांश

Statistics begins with data. This topic explains how data are collected (primary and secondary sources, census and sample methods), how raw data are classified into a frequency distribution, and how they are presented through tables, diagrams and graphs so that patterns can be seen at a glance.

महत्त्वाचे मुद्दे

• Statistics (सांख्यिकी) in the plural means numerical facts; in the singular, the science of collecting, organising, presenting, analysing and interpreting data.
• Primary data: collected first-hand for the purpose (original). Methods: direct personal interview, indirect oral investigation, questionnaire by post/e-mail/online form, schedules filled by enumerators, information through local correspondents.
• Secondary data: already collected by others (published reports, Census of India, RBI bulletins, journals, websites). Check reliability, suitability and adequacy before use.
• Census method: every item of the population is studied. Sample method: only a representative part is studied; it saves time and cost.
• Sampling methods: random (lottery, random number tables), stratified, systematic, cluster, and non-random (judgement, quota, convenience).
• Classification: arranging data into groups by common characteristics – geographical, chronological, qualitative, quantitative.
• A frequency distribution groups values into class intervals with their frequencies.
• Exclusive classes (0–10, 10–20: upper limit excluded) vs inclusive classes (0–9, 10–19).
• Tabulation: presenting data in rows and columns. Parts of a table: table number, title, caption (column heads), stub (row heads), body, head note, footnote, source.
• Diagrams: simple, multiple, subdivided and percentage bar diagrams; pie diagram. Graphs: histogram, frequency polygon, frequency curve, ogives (less-than and more-than cumulative frequency curves).

महत्त्वाच्या संज्ञा

• Population (universe) – all items under study.
• Sample – a part of the population selected for study.
• Class interval – the range of a class (upper limit − lower limit).
• Class mark (mid-value) – (lower limit + upper limit) ÷ 2.
• Cumulative frequency – running total of frequencies.
• Ogive – graph of cumulative frequencies; used to find median graphically.

नमुने व सूत्रे

• Class width = Upper limit − Lower limit
• Mid-value = (Lower limit + Upper limit) ÷ 2
• Pie diagram angle for a part = (Part ÷ Total) × 360°

Worked example. Marks of 20 students: 12, 25, 37, 8, 45, 22, 31, 18, 27, 39, 15, 28, 33, 5, 41, 24, 29, 19, 35, 26. Prepare a frequency distribution with classes 0–10, 10–20, ... (exclusive).

Pie example: a family spends ₹6,000 of ₹24,000 on rent. Angle = (6,000 ÷ 24,000) × 360° = 90°.

५ महत्त्वाचे प्रश्न

Q1. Distinguish between primary and secondary data.

Primary data are collected for the first time by the investigator for a specific purpose; they are original but costly and slow. Secondary data were collected earlier by someone else and are used again; they are cheap and quick but may not exactly suit the purpose.

Q2. What is a sample? State two merits of the sample method.

A sample is a small representative part of the population selected for study. Merits: it saves time, money and labour, and it is the only possible method when testing destroys the item (e.g. testing bulbs or medicines).

Q3. Explain the methods of collecting primary data.

Outline: direct personal investigation → indirect oral investigation → information from local agents/correspondents → mailed questionnaire → schedules through enumerators → online surveys → merits, demerits and suitability of each → qualities of a good questionnaire.

Q4. What is tabulation? Explain the parts of a good statistical table.

Outline: meaning and objectives → parts (number, title, head note, captions, stubs, body, footnote, source) with a small specimen table → rules of tabulation (clear title, units mentioned, totals, comparisons side by side).

Q5. Prepare a frequency distribution and draw a histogram and less-than ogive (numerical).

Steps: find range → decide classes (5–10) → tally marks → frequencies → cumulative frequencies → histogram: class limits on x-axis, frequency on y-axis, adjacent bars → ogive: plot upper limits against less-than cumulative frequencies and join smoothly (see the worked example).

परीक्षेसाठी टीप

Always write the title, units and source with any table or diagram. These small items carry marks and are often forgotten.

थोडक्यात सारांश

A measure of central tendency (average) is a single value that represents the whole data set. The main averages are the arithmetic mean, median and mode; geometric and harmonic means are used for special cases like growth rates and speeds. This topic gives their formulas for individual, discrete and continuous series, with a fully worked grouped-data example.

महत्त्वाचे मुद्दे

• An average (सरासरी / मध्यवर्ती प्रवृत्ती) should be rigidly defined, easy to understand, based on all values, not unduly affected by extreme values, and capable of further algebraic treatment.
• Arithmetic mean: sum of values ÷ number of values; based on all items but affected by extreme values.
• Median: the middle value when data are arranged in order; not affected by extreme values; useful for income, wages and qualitative data.
• Mode: the value that occurs most often; useful for sizes of shoes and clothes, and the most popular item.
• Weighted mean: used when items have different importance (index numbers, average marks with different credits).
• Geometric mean: average growth rate; harmonic mean: average speed or rate.
• Quartiles (Q1, Q3), deciles and percentiles divide data into 4, 10 and 100 parts.
• Empirical relation for moderately skewed data: Mode ≈ 3 Median − 2 Mean.
• Median can be found graphically from the intersection of the less-than and more-than ogives; mode from a histogram.

महत्त्वाच्या संज्ञा

• Median class – class where the cumulative frequency first reaches N/2.
• Modal class – class with the highest frequency.
• Step deviation – (x − A) ÷ h, used to simplify mean calculation.
• Bimodal – data with two modes.

नमुने व सूत्रे

• Mean (individual): x̄ = Σx ÷ n; (discrete/continuous): x̄ = Σfx ÷ N, where x = mid-value
• Step deviation: x̄ = A + (Σfd' ÷ N) × h, where d' = (x − A)/h
• Median (continuous): Md = L + [(N/2 − cf) ÷ f] × h
• Mode (continuous): Mo = L + [(f₁ − f₀) ÷ (2f₁ − f₀ − f₂)] × h
• L = lower limit of the class; cf = cumulative frequency before the class; f = frequency of the class; f₁ = modal class frequency, f₀ and f₂ = frequencies of the classes before and after; h = class width
• GM = (x₁ × x₂ × ... × xₙ)^(1/n); HM = n ÷ Σ(1/x)

Worked example (N = 40):

• Mean = 1,020 ÷ 40 = 25.5
• Median: N/2 = 20 → median class 20–30 (L = 20, cf = 13, f = 12, h = 10). Md = 20 + [(20 − 13) ÷ 12] × 10 = 20 + 5.83 = 25.83
• Mode: modal class 20–30 (f₁ = 12, f₀ = 8, f₂ = 10). Mo = 20 + [(12 − 8) ÷ (24 − 8 − 10)] × 10 = 20 + (4 ÷ 6) × 10 = 26.67

Check: 3 × 25.83 − 2 × 25.5 = 26.5, close to 26.67, as expected for nearly symmetric data.

५ महत्त्वाचे प्रश्न

Q1. What are the qualities of a good average?

It should be rigidly defined, simple to understand and calculate, based on all observations, least affected by extreme values and sampling fluctuations, and suitable for further algebraic treatment. The arithmetic mean satisfies most of these.

Q2. Find the median of 7, 3, 9, 12, 5, 10.

Arrange: 3, 5, 7, 9, 10, 12 (n = 6, even). Median = average of the 3rd and 4th items = (7 + 9) ÷ 2 = 8.

Q3. Calculate the mean, median and mode of a grouped frequency distribution (numerical).

Steps: mid-values → fx → mean → cumulative frequency → median class and formula → modal class and formula → interpret (see the worked example).

Q4. Compare mean, median and mode: merits, demerits and uses.

Outline: definitions → merits and demerits of each → uses (mean for marks and output; median for income, open-end classes; mode for fashion sizes and popular items) → empirical relation.

Q5. A vehicle travels 60 km at 30 km/h and returns at 60 km/h. Find the average speed.

Equal distances, so use the harmonic mean: HM = 2 ÷ (1/30 + 1/60) = 2 ÷ (3/60) = 40 km/h. (Check: total time = 2 h + 1 h = 3 h for 120 km = 40 km/h.)

परीक्षेसाठी टीप

Write the values of L, cf, f, f₀, f₂ and h separately before putting them in the formula. It avoids slips and earns step marks.

थोडक्यात सारांश

Two data sets can have the same average but very different spread. Dispersion (variation) measures how far the values are scattered around the average. This topic covers absolute measures (range, quartile deviation, mean deviation, standard deviation) and relative measures (their coefficients and the coefficient of variation), which are used to compare the consistency of two or more series.

महत्त्वाचे मुद्दे

• Dispersion (अपस्करण / विचलन) shows the reliability of an average: less dispersion means the average represents the data better.
• Absolute measures are in the original units (₹, kg); relative measures are pure numbers (ratios or percentages) used for comparison.
• Range = largest − smallest value; simplest but depends only on two extreme items.
• Quartile deviation (semi-interquartile range) = (Q3 − Q1)/2; ignores the top and bottom 25%, so it is not affected by extreme values; useful for open-end classes.
• Mean deviation: average of absolute deviations from mean or median (signs ignored).
• Standard deviation (σ) (प्रमाण विचलन): square root of the mean of squared deviations from the arithmetic mean; the best and most widely used measure (Karl Pearson, 1893).
• Variance = σ².
• Coefficient of variation (CV) = (σ ÷ mean) × 100. A series with lower CV is more consistent, uniform or stable.
• SD is not changed by adding a constant to every value (change of origin), but it is changed by multiplying (change of scale).
• Skewness measures lack of symmetry; kurtosis measures peakedness (introductory).

महत्त्वाच्या संज्ञा

• Consistency – less variation; lower CV.
• Q1, Q3 – values below which 25% and 75% of observations lie.
• Absolute deviation – deviation with the sign ignored, |x − x̄|.
• Karl Pearson's coefficient of skewness = (Mean − Mode) ÷ σ.

नमुने व सूत्रे

• Range = H − L; Coefficient of range = (H − L) ÷ (H + L)
• QD = (Q3 − Q1) ÷ 2; Coefficient of QD = (Q3 − Q1) ÷ (Q3 + Q1)
• Mean deviation = Σ|x − x̄| ÷ n (or Σf|x − x̄| ÷ N)
• σ = √[Σ(x − x̄)² ÷ n]; grouped: σ = √[Σf(x − x̄)² ÷ N]
• Shortcut: σ = √[Σfd² ÷ N − (Σfd ÷ N)²], d = x − A
• CV = (σ ÷ x̄) × 100

Worked example: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8, mean = 40 ÷ 8 = 5).

• Range = 9 − 2 = 7; coefficient = 7 ÷ 11 = 0.64
• Mean deviation = 12 ÷ 8 = 1.5
• σ = √(32 ÷ 8) = √4 = 2; variance = 4
• CV = (2 ÷ 5) × 100 = 40%
• Q1 = value of the (n + 1)/4 = 2.25th item = 4; Q3 = 6.75th item = 5 + 0.75 × (7 − 5) = 6.5; QD = (6.5 − 4) ÷ 2 = 1.25

५ महत्त्वाचे प्रश्न

Q1. What is the coefficient of variation? Why is it used?

CV = (σ ÷ mean) × 100. It is a relative measure of dispersion used to compare the variability or consistency of two or more series, even when their units or averages differ. The series with the lower CV is more consistent.

Q2. Two workers' daily output: A has mean 50 units, σ 5; B has mean 40 units, σ 6. Who is more consistent?

CV of A = (5 ÷ 50) × 100 = 10%. CV of B = (6 ÷ 40) × 100 = 15%. A has the lower CV, so A is more consistent.

Q3. Calculate the standard deviation of a grouped frequency distribution (numerical).

Steps: mid-values → mean → deviations → squared deviations × f → σ = √[Σf(x − x̄)² ÷ N]. Example: classes 0–10, 10–20, 20–30, 30–40, 40–50 with f = 5, 8, 12, 10, 5 give mean 25.5, Σf(x − x̄)² = 5,790, variance 144.75, σ ≈ 12.03 and CV ≈ 47.2%.

Q4. Compare the various measures of dispersion.

Outline: range, QD, MD, SD with definition, formula, merits and demerits → why SD is the best (based on all values, algebraic treatment, used in sampling and correlation) → when QD is preferred (open-end classes).

Q5. Explain the meaning and objectives of measuring dispersion.

Outline: meaning → objectives (test reliability of average, compare consistency, control quality, basis for further analysis like correlation and regression) → properties of a good measure → absolute vs relative measures.

परीक्षेसाठी टीप

For comparison questions, never compare σ alone when the means differ; always compute the CV and state which series is more consistent.

थोडक्यात सारांश

Correlation measures the degree and direction of the relationship between two variables, such as advertising and sales or price and demand. Regression goes one step further: it gives an equation to estimate the value of one variable from a known value of the other. This topic covers Karl Pearson's coefficient, Spearman's rank correlation and the two regression lines.

महत्त्वाचे मुद्दे

• Correlation (सहसंबंध): positive (both move in the same direction: income and spending), negative (opposite directions: price and demand), or zero (no relation).
• Linear vs non-linear; simple (two variables) vs multiple and partial correlation.
• Methods: scatter diagram, Karl Pearson's coefficient (r), Spearman's rank correlation (ρ), concurrent deviation method.
• r lies between −1 and +1: +1 perfect positive, −1 perfect negative, 0 no linear correlation.
• r is independent of change of origin and scale.
• Correlation does not prove cause and effect; two variables can be correlated by chance or through a third factor.
• Spearman's rank method is used for qualitative data (beauty, honesty) or when only ranks are given.
• Regression (प्रतीपगमन): there are two regression lines – Y on X (estimate Y from X) and X on Y (estimate X from Y). Both pass through the point (x̄, ȳ).
• Regression coefficients: byx and bxy; r = ±√(byx × bxy), with the same sign as the coefficients. Both coefficients have the same sign, and both cannot be greater than 1.
• If r = ±1, the two regression lines coincide; if r = 0, they are perpendicular.

महत्त्वाच्या संज्ञा

• Scatter diagram – plot of paired values to see the pattern of relationship.
• Covariance – average of the products of deviations: Σ(x − x̄)(y − ȳ) ÷ n.
• Probable error – PE = 0.6745 × (1 − r²) ÷ √n; if r > 6 PE, correlation is significant.
• Regression coefficient (byx) – change in Y for one unit change in X.

नमुने व सूत्रे

• r = Σxy ÷ √(Σx² × Σy²), where x = X − X̄ and y = Y − Ȳ (deviations from means)
• Spearman: ρ = 1 − [6Σd² ÷ n(n² − 1)], d = difference in ranks
• byx = Σxy ÷ Σx²; bxy = Σxy ÷ Σy²; also byx = r × σy/σx
• Regression of Y on X: Y − Ȳ = byx (X − X̄)
• Regression of X on Y: X − X̄ = bxy (Y − Ȳ)

Worked example: X = 1, 2, 3, 4, 5 (advertising, ₹ thousand); Y = 2, 4, 5, 4, 5 (sales, ₹ lakh). X̄ = 3, Ȳ = 4.

• r = 6 ÷ √(10 × 6) = 6 ÷ 7.746 = 0.77 (high positive correlation)
• byx = 6 ÷ 10 = 0.6 → Y − 4 = 0.6(X − 3) → Y = 0.6X + 2.2
• bxy = 6 ÷ 6 = 1 → X − 3 = 1(Y − 4) → X = Y − 1
• Check: √(0.6 × 1) = 0.7746 = r
• Estimate: if advertising X = 6, sales Y = 0.6 × 6 + 2.2 = 5.8

५ महत्त्वाचे प्रश्न

Q1. What is the meaning of r = −0.9?

It shows a high degree of negative correlation: when one variable increases, the other decreases strongly and almost linearly, e.g. price and quantity demanded.

Q2. Find Spearman's rank correlation if two judges rank 5 candidates as (1, 2, 3, 4, 5) and (2, 1, 4, 3, 5).

d = −1, 1, −1, 1, 0; Σd² = 4. ρ = 1 − (6 × 4) ÷ [5 × (25 − 1)] = 1 − 24/120 = 0.8. There is a high positive agreement between the judges.

Q3. Calculate Karl Pearson's coefficient of correlation (numerical).

Steps: find means → deviations x and y → x², y², xy → totals → apply formula → interpret (see worked example). For large values use assumed means with the formula r = [nΣdxdy − ΣdxΣdy] ÷ √{[nΣdx² − (Σdx)²][nΣdy² − (Σdy)²]}.

Q4. Obtain the two regression equations and estimate Y when X = 6 (numerical).

Steps: means → byx and bxy → write both equations through (x̄, ȳ) → simplify → substitute X = 6 in the Y-on-X line → check that r² = byx × bxy (see worked example: Y = 0.6X + 2.2, X = Y − 1, estimate 5.8).

Q5. Distinguish between correlation and regression.

Outline: correlation measures degree and direction; regression gives a functional relation for estimation → correlation is symmetric (rxy = ryx); regression has two different lines → correlation does not show cause and effect; regression treats one variable as dependent → r is a pure number; regression coefficients have units → uses in business forecasting.

परीक्षेसाठी टीप

Always use the Y-on-X line to estimate Y and the X-on-Y line to estimate X. Using the wrong line is the commonest error in regression problems.