SSC Statistics Notes: Complete Study Material for CGL Tier 2 (JSO)
Statistics is the specialised subject of Paper II of SSC CGL Tier 2, which is meant for the candidates who want the post of Junior Statistical Officer (JSO) in the Ministry of Statistics and Programme Implementation. Candidates who apply for the other CGL posts do not have to attempt this paper. Statistics is a scoring subject because the syllabus is fixed, the formulas are standard and the questions are mostly numerical and direct. A student who understands the meaning of each measure, and not only its formula, can solve most questions in less than a minute.
These SSC Statistics notes are written in simple language for students who are studying the subject for the first time as well as for graduates who need a quick revision. Every chapter below gives the main definitions, the important formulas and the points that the examiner likes to ask. Use this page as a map of the syllabus, and then practise the numerical questions of each chapter.
What You Get in These Statistics Notes
- A chapter-wise map of the complete SSC CGL Tier 2 Statistics syllabus.
- All the important formulas of averages, dispersion, correlation, regression, probability and distributions in one place.
- Short explanations of the terms that are asked as theory questions, such as skewness, kurtosis, standard error and the types of errors in hypothesis testing.
- A preparation plan, the exam tips and the links to related subjects like Quantitative Aptitude for the basic maths.
Exam Pattern and Syllabus Overview
| Point | Detail |
|---|---|
| Exam and tier | SSC CGL Tier 2 |
| Paper | Paper II: Statistics |
| Post | Junior Statistical Officer (JSO) |
| Mode | Computer-based test |
| Type of questions | Objective multiple choice questions |
| Usual size | About 100 questions of 2 marks each (200 marks) in 2 hours, with negative marking |
| Main areas | Descriptive statistics, probability, distributions, sampling, inference, time series and index numbers |
The number of questions, the marks and the negative marking can change in a new notification, so always check the latest official SSC notice before the exam.
Collection, Classification and Presentation of Data
Data are the facts and figures collected for a study. Primary data are collected first-hand by the investigator (by a survey or a questionnaire), while secondary data are taken from published sources such as the census and government reports. After collection, the data are classified. In a frequency distribution, the values are grouped into class intervals and the number of observations in each class is its frequency.
- Class width (class size) = upper limit - lower limit of the class (for continuous classes).
- Class mark (mid-value) = (lower limit + upper limit) / 2.
- Cumulative frequency is the running total of the frequencies. It is used to find the median and the quartiles.
- Relative frequency = class frequency / total frequency.
- Presentation of data: tables, bar diagrams, pie charts, histogram (for continuous data), frequency polygon, frequency curve and ogive (the graph of cumulative frequency, used to find the median).
Measures of Central Tendency
A measure of central tendency gives one value that represents the whole data. The main measures are the mean, the median, the mode, the geometric mean and the harmonic mean.
| Measure | Formula and key points |
|---|---|
| Arithmetic mean (AM) | Sum of observations / number of observations. For a frequency table: sum of (f x x) / sum of f. The sum of the deviations of the items from their mean is zero. |
| Median | The middle value of the ordered data. For a grouped series: L + [(N/2 - cf) / f] x h, where L is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h its width. |
| Mode | The most frequent value. For a grouped series: L + [(f1 - f0) / (2f1 - f0 - f2)] x h, where f1 is the frequency of the modal class and f0, f2 are the frequencies before and after it. |
| Geometric mean (GM) | The nth root of the product of n values. Used for rates of growth and index numbers. |
| Harmonic mean (HM) | n / sum of (1/x). Used for averaging speeds and rates. |
- For positive values, AM is greater than or equal to GM, and GM is greater than or equal to HM. They are equal only when all the values are equal.
- For two positive numbers a and b: GM squared = AM x HM.
- For a moderately skewed distribution: Mode = 3 Median - 2 Mean.
- The mean is affected most by extreme values, the median is not, and the mode can be found for qualitative data also.
- Quartiles divide the data into 4 equal parts, deciles into 10 parts and percentiles into 100 parts. The median is Q2.
Measures of Dispersion
Dispersion shows how far the values are spread around the centre. Two data sets can have the same mean but different spread, so the dispersion is as important as the average.
| Measure | Formula |
|---|---|
| Range | Largest value - smallest value |
| Quartile deviation (QD) | (Q3 - Q1) / 2 |
| Mean deviation | Sum of |x - average| / n (the average is the mean or the median) |
| Variance | Sum of (x - mean) squared / n |
| Standard deviation (SD) | Square root of the variance |
| Coefficient of variation (CV) | (SD / mean) x 100; the lower CV means the more consistent data |
| Coefficient of range | (Largest - smallest) / (largest + smallest) |
| Coefficient of QD | (Q3 - Q1) / (Q3 + Q1) |
- SD is the best and the most used measure of dispersion.
- If every value is increased by a constant, the SD does not change; if every value is multiplied by k, the SD is multiplied by the absolute value of k.
- Variance of the first n natural numbers = (n squared - 1) / 12.
- The sum of the squares of the deviations is the least when taken from the mean.
Moments, Skewness and Kurtosis
- Moments are the averages of the powers of the deviations. The first central moment is zero and the second central moment is the variance.
- Skewness shows the lack of symmetry. In a symmetric distribution, mean = median = mode. In a positively skewed distribution, the tail is on the right and mean > median > mode; in a negatively skewed distribution, the tail is on the left and mean < median < mode.
- Karl Pearson's coefficient of skewness = (Mean - Mode) / SD. Bowley's coefficient = (Q3 + Q1 - 2 Median) / (Q3 - Q1), which lies between -1 and +1.
- Kurtosis shows the peakedness of the curve. A normal curve is mesokurtic (beta 2 = 3), a more peaked curve is leptokurtic (beta 2 greater than 3) and a flatter curve is platykurtic (beta 2 less than 3).
Correlation and Regression
Correlation measures the degree and the direction of the linear relation between two variables. The correlation coefficient r always lies between -1 and +1. A value near +1 shows a strong positive relation, near -1 a strong negative relation and near 0 almost no linear relation.
- Karl Pearson's coefficient: r = Cov(x, y) / (SD of x x SD of y). It is not affected by a change of origin or scale (with positive scale).
- Spearman's rank correlation: r = 1 - [6 x sum of d squared] / [n (n squared - 1)], where d is the difference of the ranks.
- Regression is used to estimate one variable from the other. The regression line of y on x is y - (mean of y) = b(yx) (x - mean of x), where b(yx) = r x (SD of y / SD of x). The regression line of x on y uses b(xy) = r x (SD of x / SD of y).
- Both regression lines pass through the point (mean of x, mean of y).
- r squared = b(yx) x b(xy), so r is the geometric mean of the two regression coefficients, and both coefficients have the same sign as r.
- If r = 0 the two regression lines are at right angles; if r = +1 or -1 the two lines coincide.
Probability
- Probability of an event = number of favourable outcomes / total number of outcomes. It lies between 0 and 1.
- Addition theorem: P(A or B) = P(A) + P(B) - P(A and B). For mutually exclusive events, P(A and B) = 0.
- Multiplication theorem: P(A and B) = P(A) x P(B given A). For independent events, P(A and B) = P(A) x P(B).
- Conditional probability: P(B given A) = P(A and B) / P(A).
- Bayes' theorem finds the probability of a cause when the effect has been observed.
- The probability of the complement of an event A = 1 - P(A).
- Expected value of a random variable = sum of x x P(x).
Probability Distributions
| Distribution | Key facts |
|---|---|
| Binomial | n trials, each with success probability p and failure probability q = 1 - p. P(X = r) = nCr x p^r x q^(n - r). Mean = np and variance = npq. |
| Poisson | For rare events in a fixed interval. P(X = r) = e^(-m) x m^r / r!. Mean = variance = m. |
| Normal | A bell-shaped symmetric curve. Mean = median = mode. About 68 percent of the values lie within 1 SD, 95 percent within 2 SD and 99.7 percent within 3 SD of the mean. Standard normal variable Z = (x - mean) / SD. |
| Uniform | All the values in a range are equally likely. |
Sampling Theory and Statistical Inference
- A population is the whole group under study and a sample is a part of it. A number that describes a population is a parameter, and one that describes a sample is a statistic.
- Sampling methods: simple random sampling, stratified sampling, systematic sampling and cluster sampling. Random sampling gives every unit an equal chance.
- Sampling error arises because only a part of the population is studied, while non-sampling errors arise from wrong data collection and processing.
- The standard error of the sample mean = SD / square root of n.
- Hypothesis testing: the null hypothesis (H0) is the statement tested and the alternative hypothesis (H1) is the opposite. A Type I error is rejecting a true H0 and a Type II error is accepting a false H0. The level of significance is the probability of the Type I error (usually 5 percent or 1 percent).
- Common tests: the Z-test (large samples), the t-test (small samples), the chi-square test (for fit and independence) and the F-test (for comparing variances).
- Analysis of variance (ANOVA) compares the means of more than two groups using the F-test.
Time Series and Index Numbers
- A time series is a set of observations arranged in time. Its four components are secular trend, seasonal variation, cyclical variation and irregular variation.
- Methods of finding the trend: free-hand curve, semi-averages, moving averages and the method of least squares.
- An index number measures the change in the level of a variable over time. Laspeyres' index uses the base-year quantities as weights, Paasche's index uses the current-year quantities and Fisher's ideal index is the geometric mean of the two.
- The Consumer Price Index (CPI) measures the change in the retail prices of a basket of goods and services and is used to measure inflation. The Wholesale Price Index (WPI) measures the change in the wholesale prices.
- Time reversal test and factor reversal test are the tests of an ideal index number; Fisher's index satisfies both.
How to Prepare SSC Statistics
- Study the chapters in the order given above, because the later chapters depend on the earlier ones.
- Prepare a formula sheet of one page for every chapter and revise it every week.
- Solve at least 20 numerical questions after each chapter, with a timer of one minute per question.
- Learn the theory points (skewness, kurtosis, errors in testing) as short facts, because they come as direct questions.
- Attempt full-length mock tests in the last month and analyse the weak chapters.
- Revise the basic maths of ratio, percentage and average from the Quantitative Aptitude notes such as average and percentage.
Tips for the Exam
- Do not spend more than two minutes on any question.
- Check the units and the type of data (grouped or ungrouped) before putting the formula.
- In theory questions, eliminate the options that are clearly wrong and then choose.
Practice and Next Steps
After the theory, practise with the SSC practice papers, solve the previous year papers and try the free SSC quizzes. Because every SSC candidate also has to prepare the common subjects, keep a part of your time for Reasoning, English, General Awareness and Computer Knowledge.
Explore Other SSC Subjects
SSC exams test many subjects, so a good study plan connects them. Move to the next subject once you finish this one:
- Quantitative Aptitude Notes: Maths formulas, tricks and solved examples
- Reasoning Notes: Verbal and non-verbal reasoning topics
- English Notes: Grammar, vocabulary and comprehension
- General Awareness Notes: History, polity, geography, economy and static GK
- Computer Knowledge Notes: Computer basics, MS Office, internet and security
- General Science Notes: Physics, chemistry and biology for SSC
- Finance and Economics Notes: Finance, accounts and economics for CGL Tier 2 (AAO)
- General Hindi Notes: Hindi grammar and vocabulary
- Current Affairs Notes: Latest national and international events
- Indian History Notes: Ancient, medieval and modern India
- Indian Geography Notes: Physical, economic and world geography
- Indian Polity Notes: Constitution, parliament and governance
- Indian Economy Notes: Economy, budget, banking and schemes
- Civil Engineering (SSC JE) Notes: Civil engineering topics for SSC JE
- Mechanical Engineering (SSC JE) Notes: Mechanical engineering topics for SSC JE
- Electrical Engineering (SSC JE) Notes: Electrical engineering topics for SSC JE
- Stenography Notes: Shorthand, typing and transcription for Stenographer
You can also practise with the SSC practice papers, solve the previous year papers and attempt the free SSC quizzes.
Frequently Asked Questions About SSC Statistics Notes
For which SSC exam is the Statistics subject needed?
Statistics is the subject of Paper II of SSC CGL Tier 2, which is for the candidates applying for the post of Junior Statistical Officer (JSO).
Is Statistics compulsory for all SSC CGL candidates?
No. It is only for the candidates who want the JSO post. The other candidates attempt the papers that are required for their posts.
What are the most important topics of SSC Statistics?
The most important topics are measures of central tendency, dispersion, correlation and regression, probability, probability distributions, sampling and hypothesis testing, and index numbers.
What is the relation between mean, median and mode?
For a moderately skewed distribution, Mode = 3 Median - 2 Mean. In a symmetric distribution all three are equal.
What is the range of the correlation coefficient?
The correlation coefficient r lies between -1 and +1.
What is the difference between a parameter and a statistic?
A parameter is a number that describes a population, while a statistic is a number that describes a sample.
Which formula is used for the standard error of the mean?
The standard error of the sample mean is the standard deviation divided by the square root of the sample size.
How should I prepare Statistics for the SSC exam?
Learn the formulas chapter by chapter, practise numerical questions daily, revise theory points as short facts and attempt mock tests before the exam.
Which formulas should I learn first for SSC Statistics?
Start with the formulas of mean, median, mode, standard deviation and variance, then learn the correlation coefficient, the regression equations and the basic rules of probability.
What is the difference between correlation and regression?
Correlation measures the strength and direction of the relationship between two variables. Regression gives an equation that is used to estimate the value of one variable from the other.
How many hours should I study Statistics daily?
Give 2 to 3 hours daily for the Statistics paper. Spend the first hour on concepts and formulas and the remaining time on solving numerical questions.
How can I practise numerical questions in Statistics?
Solve at least 15 questions from each chapter, write the formula before every step and check your answer once. Then attempt the previous year papers and practice papers in a fixed time.