First Year Higher Secondary Statistics Study Guide

Definitions, Scope, and Limitations of Statistics

• Introduction to Statistics:

  • In the modern era of computers and information technology, the importance of statistics is recognized across all disciplines.

  • Originally evolved as a science of statehood, but now finds applications in Agriculture, Economics, Commerce, Biology, Medicine, Industry, Education, and Planning.

• Origin and Growth:

  • The word ‘Statistics’ is derived from the Latin word Status, which means a political state.

  • It is a comparatively recent branch of the scientific method, with continuous research into its mathematical theory globally.

• Verbatim Definition - Croxton and Cowden:

  • "Statistics may be defined as the science of collection, presentation, analysis and interpretation of numerical data from the logical analysis."

• Verbatim Definition - Horace Secrist:

  • "Statistics may be defined as the aggregate of facts affected to a marked extent by multiplicity of causes, numerically expressed, enumerated or estimated according to a reasonable standard of accuracy, collected in a systematic manner, for a predetermined purpose and placed in relation to each other."

• Bowley’s Definitions:

  • "Statistics are numerical statement of facts in any department of enquiry placed in relation to each other."

  • "Statistics may be called the science of counting."

  • "Statistics may be rightly called the scheme of averages."

  • Note: These are considered incomplete as they ignore aspects like interpretation and analysis.

• Functions of Statistics:

  • Condensation: Reducing huge masses of data into manageable observations (e.g., using averages and ranges).

  • Comparison: Using classification and tabulation to compare data across regions or sources.

  • Forecasting: Predicting future trends (e.g., rainfall, business profit) using Time Series and Regression analysis.

  • Estimation: Drawing inferences about a population from a sample (Estimation Theory).

  • Tests of Hypothesis: Formulating and testing statements about population distributions (e.g., drug efficacy).

• Scope of Statistics:

  • Industry: Uses Control Charts and inspection plans to maintain quality levels.

  • Commerce: Market surveys and demand forecasting are essential for managing stock and competition.

  • Agriculture: Analysis of Variance (ANOVA), developed by Professor R.A. Fisher, tests the significance of differences in crop yields under various fertilizers.

  • Economics: Alfred Marshall stated, "Statistics are the straw only which I like every other economist have to make the bricks."

  • Planning: Indispensable for government policy formulation regarding production, consumption, and income-expenditure.

  • Medicine: Uses the $t-\text{test}$ to compare the efficiency of different drugs.

• Limitations:

  • Qualitative Data: It cannot directly study honesty, beauty, or poverty unless reduced to numerical terms.

  • Individuals: It deals only with aggregates, not individual items.

  • Approximations: Statistical laws are not as exact as physical sciences; they are true only on average.

  • Misuse: Prof. King observes, "Statistics are like clay of which one can make a God or Devil as one pleases."

Introduction to Sampling Methods

• Population and Census:

  • Population (Universe): The complete set of observations investigated.

  • Finite Population: Consists of a reachable number of units (e.g., workers in a factory).

  • Infinite Population: Uncountable units (e.g., stars in the sky).

  • Census Method: Every element of the population is surveyed. It is accurate but costly and time-consuming.

• Sampling Terminology:

  • Sample: A finite subset of individuals chosen from the population.

  • Sample Size ($n$): The number of units in the sample.

  • Sampling Unit: The individuals to be sampled (e.g., a family head for an income survey).

  • Sampling Frame: A list or map identifying each sampling unit (e.g., voters list).

• Parameters vs. Statistics:

  • Parameters: Characteristics of a population (Mean $\mu$, Standard Deviation $\sigma$, size $N$).

  • Statistics: Characteristics of a sample (Mean $\bar{x}$, s.d. $s$, size $n$).

• Principles of Sampling:

  • Statistical Regularity: A large number of units chosen at random will likely possess the characteristic of the whole group.

  • Inertia of Large Numbers: Accuracy increases with sample size.

  • Optimisation: Achieving maximum information with minimum cost and time.

• Types of Sampling:

  • Probability (Random): Selection based on known probabilities (e.g., Simple Random Sampling).

  • Non-Probability (Non-Random): Based on personal judgment or quotas.

  • Simple Random Sampling (SRS): Every item has an equal probability. Methods include the Lottery Method and the Table of Random Numbers (Tippett’s, Fisher and Yates’, or Kendall and Smith’s tables).

  • Stratified Random Sampling: Dividing heterogeneous populations into homogeneous strata. Proportional allocation uses $\frac{n}{N} = c$.

  • Systematic Sampling: Selecting every $K^{th}$ element, where $K = \frac{N}{n}$.

Collection of Data, Classification, and Tabulation

• Nature of Data:

  • Time Series: Collected over time (e.g., household annual expenditure).

  • Spatial Data: Relates to geographical locations (e.g., district-wise rainfall).

  • Spacio-Temporal Data: Relates to both time and space (e.g., population of states across census years).

• Primary Data Collection Methods:

  • Direct Personal Interviews: High response rate and accurate, but costly.

  • Indirect Oral Interviews: Interviewing third parties (e.g., for theft or murder cases).

  • Questionnaires: A series of questions mailed to respondents. A good questionnaire should have short, simple, logical, and non-sensitive questions.

  • Schedules: Similar to questionnaires but filled by trained enumerators during face-to-face contact.

• Secondary Data:

  • Data collected by others (published reports, journals, government records).

  • Sources: IMF, UN, Central/State governments, trade bodies, business journals.

• Classification and Tabulation:

  • Classification Types: Chronological (Time), Geographical (Region), Qualitative (Attributes like sex/literacy), and Quantitative (Measurable variables like height/weight).

  • Table Structure: Table Number, Title, Captions (vertical headings), Stubs (horizontal headings), Body, Footnotes, and Source.

Frequency Distribution

• Sturges' Rule:

  • To determine the number of class intervals ($K$): $K = 1 + 3.322\log_{10}(N)$.

  • The width of class interval ($C$): $C = \frac{\text{Range}}{1 + 3.322\log_{10}(N)}$.

• Discrete vs. Continuous:

  • Discrete: Variables with definite differences (e.g., number of children).

  • Continuous: Variables take any fractional value (e.g., weights in kgs).

• Classification Methods:

  • Exclusive Method: Upper limit of one class is the lower limit of the next (e.g., 0-10, 10-20).

  • Inclusive Method: Both limits are included (e.g., 10-19, 20-29).

  • Open-end Classes: Limits missing at the start or end (e.g., "Below 2000" or "Above 8000").

• Cumulative Frequencies:

  • Less than: Running total from top down.

  • More than: Running total from bottom up.

Diagrammatic and Graphical Representation

• One-Dimensional Diagrams:

  • Line, Simple Bar, Multiple Bar (comparing subsets), Sub-divided Bar (components), and Percentage Bar.

• Area/Two-Dimensional Diagrams:

  • Rectangles, Squares, and Pie Diagrams (Sector calculation: $\frac{\text{Component Value}}{\text{Total}} \times 360^{\circ}$).

• Graphs:

  • Histogram: Rectangles representing frequencies; width is class interval.

  • Frequency Polygon: Midpoints of histogram rectangles joined by straight lines.

  • Frequency Curve: Smooth freehand curve through the polygon points.

  • Ogive: Cumulative frequency curve (used to find Median).

  • Lorenz Curve: Measures socio-economic inequality (e.g., wealth distribution).

Measures of Central Tendency

• Arithmetic Mean ($\bar{x}$):

  • Ungrouped: $\bar{x} = \frac{\sum x}{n}$.

  • Grouped (Assumed Mean Method): $\bar{x} = A + \frac{\sum fd}{N} \times c$, where $d = \frac{x-A}{c}$.

  • Weighted Mean: $\bar{x}_{w} = \frac{\sum wx}{\sum w}$.

• Geometric Mean (G.M.):

  • Defined as the $n^{th}$ root of the product of $n$ observations.

  • Formula: $\text{G.M.} = \text{Antilog}\left( \frac{\sum \log x}{n} \right)$.

• Harmonic Mean (H.M.):

  • Reciprocal of the arithmetic average of the reciprocals of observations.

  • Formula: $\text{H.M.} = \frac{n}{\sum \frac{1}{x}}$.

• Median ($M_d$):

  • The middle value dividing the distribution into two equal parts.

  • Continuous distribution formula: $M_d = L + \frac{\frac{N}{2} - m}{f} \times c$.

• Mode ($M_0$):

  • The most frequent value.

  • Formula: $M_0 = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times c$.

  • Empirical Relationship: $\text{Mode} = 3\text{Median} - 2\text{Mean}$.

Measures of Dispersion, Skewness, and Kurtosis

• Absolute Measures:

  • Range: $L - S$.

  • Quartile Deviation (Q.D.): $\frac{Q_3 - Q_1}{2}$.

  • Mean Deviation (M.D.): $\frac{\sum f|D|}{N}$.

  • Standard Deviation (S.D. or $\sigma$): $\sigma = \sqrt{\frac{\sum x^2}{n} - \left(\frac{\sum x}{n}\right)^2}$.

• Relative Measures:

  • Coefficient of Variation (C.V.): $\frac{\sigma}{\bar{x}} \times 100$. Lower C.V. means higher consistency.

• Moments:

  • Arithmetic mean of various powers of deviations from the actual mean ($\mu_r$).

  • $\mu_1 = 0$, $\mu_2 = \text{Variance}$.

• Skewness:

  • Lack of symmetry.

  • Symmetrical: $\text{Mean} = \text{Median} = \text{Mode}$.

  • Positive: $\text{Mean} > \text{Median} > \text{Mode}$.

  • Negative: $\text{Mode} > \text{Median} > \text{Mean}$.

• Kurtosis:

  • Measures peakedness.

  • Mesokurtic: $\beta_2 = 3$.

  • Leptokurtic: $\beta_2 > 3$.

  • Platykurtic: $\beta_2 < 3$.

Correlation and Regression

• Correlation Coefficient ($r$):

  • Varies between $-1$ and $+1$.

  • Karl Pearson’s Formula: $r = \frac{\text{Cov}(x,y)}{\sigma_x \sigma_y}$.

  • Spearman’s Rank Correlation: $r_s = 1 - \frac{6\sum D^2}{n(n^2-1)}$.

• Regression Analysis:

  • Predicting a dependent variable ($Y$) based on an independent variable ($X$).

  • Regression Line of $Y$ on $X$: $Y - \bar{Y} = b_{yx}(X - \bar{X})$, where $b_{yx} = r \frac{\sigma_y}{\sigma_x}$.

  • Regression Line of $X$ on $Y$: $X - \bar{X} = b_{xy}(Y - \bar{Y})$, where $b_{xy} = r \frac{\sigma_x}{\sigma_y}$.

  • Geometric Property: $r = \sqrt{b_{yx} \times b_{xy}}$.

Index Numbers

• Classification:

  • Price Index: Measures change in price level.

  • Quantity Index: Measures volume of production/consumption.

• Weighted Aggregate Indices:

  • Laspeyre’s: Uses base year weights ($p_0q_0$). $P_{01}^L = \frac{\sum p_1q_0}{\sum p_0q_0} \times 100$.

  • Paasche’s: Uses current year weights ($q_1$). $P_{01}^P = \frac{\sum p_1q_1}{\sum p_0q_1} \times 100$.

  • Fisher’s Ideal Index: Geometric mean of Laspeyre and Paasche. $P_{01}^F = \sqrt{L \times P} \times 100$.

• Tests of Consistency:

  • Time Reversal Test: $P_{01} \times P_{10} = 1$.

  • Factor Reversal Test: $P_{01} \times Q_{01} = \text{Value Ratio}$.

  • Note: Fisher’s Ideal Index satisfies both tests.

• Consumer Price Index (Cost of Living):

  • Aggregate Expenditure Method: Identical to Laspeyre’s.

  • Family Budget Method: $\frac{\sum PW}{\sum W}$, where $P = \text{Price Relative}$.