📊𝐒𝐭𝐚𝐭𝐢𝐬𝐭𝐢𝐜𝐬_𝐈𝐧𝐝𝐢𝐚🇮🇳

Coming soon. The Trial Index of Services Production (ISP) for June 2026 offering timely insights into services output and complementing the IIP while also strengthening economic monitoring, are set for release on 31 August 2026. #ISP #MoSPI #GoIStats #ServicesEconomy #IndiaEconomy #EconomicMonitoring #OfficialStatistics #DataForDevelopment #GSTData #EconomicIndicators

શબ્દો કદાચ ઝૂઠા હોઈ શકે, પણ કાર્યો હંમેશાં સત્યતા જ બતાવે છે. ​પોતાની જાત પ્રત્યે પ્રામાણિક રહો. તમે આખી દુનિયાને હજારો બહાના આપી શકો છો, પરંતુ હકીકતમાં તો તમે પોતે પણ જાણો છો કે તમે શું કરી રહ્યા છો અને તમારી ક્યાં ભૂલ થઈ રહી છે. ​દુનિયાને તમારી સફળતા, નિષ્ફળતા કે તમારી સખત મહેનત સાથે કોઈ લેવાદેવા નથી. તે માત્ર એક શાંત દર્શક છે, પરંતુ જો કંઈક ખોટું થાય તો તે તમારી ટીકા કરવા માટે હંમેશાં તૈયાર રહે છે અને જ્યારે એવું બને છે, ત્યારે ક્યાં ભૂલ થઈ તેનું આત્મનિરીક્ષણ કરવાના બદલે, આપણે દુનિયાને મનાવવા માટેના વિકલ્પો શોધવા લાગીએ છીએ. તમે દુનિયાને તો સરળતાથી મનાવી લેશો પણ તમારું મન તેના માટે અનેક બહાના પણ શોધી લેશે, પરંતુ તમે તમારી પોતાની જાતને આપેલા વચનો સાથે કેવી રીતે ન્યાય કરશો? ​તમે એક ઉત્તમ ભવિષ્ય માટે સખત મહેનત કરવાનો નિર્ણય લીધો હતો, દુનિયા માટે નહીં. તમે વધુ મહેનત કરવાની પ્રતિજ્ઞા લીધી હતી. કોઈની વાહવાહી કે સ્વીકૃતિ મેળવવા માટે નહીં, પરંતુ પોતાના એક એવા સ્વરૂપને ઘડવા માટે જેના પર તમને ખુદ ગર્વ થઈ શકે. ​તેથી પોતાની જાત સાથે વફાદાર રહો... તમારા સત્યને તમારાથી વધારે કોઈ નથી જાણતું. તેનાથી ભાગવાને બદલે આત્મનિરીક્ષણ કરો અને સુધારો લાવો, દુનિયા માટે નહીં કે દુનિયા સામે સાબિત કરવા માટે નહિ, પરંતુ તમારા પોતાના માટે. પોતાના વચનો પાળવાનું શરૂ કરો અને પોતાની જાત માટે ઊભા રહો. સફળતા મેળવો, અને મળશે જ ચોક્કસ પણે ​તમારી જાત પ્રત્યે પ્રામાણિક રહો અને પોતાને આપેલા વચનો પૂરા કરો. તમારો આજનો દિવસ મંગલમય રહે! શુભ રક્ષાબંધન 📿🎁 આવા વિચારો વાંચવા અહીં ક્લિક કરો https://t.me/statistics_india ✅ જો વિચાર ગમે તો તમારા મિત્ર વર્તુળમાં અવશ્ય શેર કરશો.

📊Normalization & Standardization in Statistics 🎯Competitive Exam • GPSC/GSS • Statistics Quick Notes Core idea: Both are data transformation techniques, but they are not the same thing. 1️⃣🔄Normalization 📌Meaning Normalization transforms numerical data to a common bounded scale, commonly 0 to 1, while preserving relative ordering. ⭐Min–Max Normalization X'= (x-xmin)/(xmax-xmin) 🔑Uses 🤖Machine Learning 📊Comparing variables with different units 📈Data preprocessing ⚙️Optimization algorithms 2️⃣📐Standardization 📌 Meaning Standardization transforms data so that the resulting variable has: Mean =0; SD=1 The most common method is the Z-score transformation. X=(x-mean)/SD 🧠Easy Memory Trick 🔄NORMALIZATION “Bring values into a common RANGE.” 0-->1 📐STANDARDIZATION “Convert values into STANDARD DEVIATIONS.” X=(x-mean)/SD 📌Min–Max → Normalization 📌Z-score → Standardization 📍Normalization # Standardization #Statistics #Normalization #Standardization #GPSC #GSS #StatisticsIndia https://youtube/statistics_india

🚨 CORRELATION: ONE NUMBER THAT CAN TELL A BIG STORY — IF YOU KNOW HOW TO READ IT! 📊🔗 When two variables move together, statisticians often want to know: 👉 How strong is the relationship? 👉 Which direction does it move? 👉 Is the relationship actually linear? That’s where correlation comes in. 🧠 WHAT IS CORRELATION? Correlation measures the strength and direction of a linear relationship between two quantitative variables. The familiar Pearson correlation coefficient, \(r\) ranges from: −1 ⟵ 0 ⟶ +1 🔴 r=-1 → Perfect negative linear relationship 🟣 r ≈ 0 → Little or no linear relationship 🟢 r=+1 → Perfect positive linear relationship 📈 BUT HERE'S THE TRAP! A strong correlation does NOT automatically mean: ❌ One variable causes the other ❌ The relationship is nonlinear-free ❌ There are no outliers ❌ The model will predict perfectly Correlation ≠ Causation. 🚨 A single influential outlier can also dramatically change Pearson's \(r\). 🔍 BEFORE INTERPRETING \(r\) 📊 Plot a scatterplot 🔎 Check for outliers 📈 Examine whether the relationship is approximately linear 🧠 Interpret the result within the real-world context 💡 STATISTICAL GOLD: > Don't just calculate the correlation coefficient. Understand the data that produced it. 🎯 QUICK QUIZ! A study finds \(r=0.82\) between study hours and examination scores. What does this indicate? 🅰️ Studying causes higher scores with certainty 🅱️ There is a strong positive linear association 🅲️ 82% of scores are caused by studying 🅳️ The relationship must be perfect 👇 Drop your answer in the comments! 📌 SAVE this for your statistics revision. 🔄 SHARE it with a statistics student, researcher, or data analyst. 👍 LIKE & FOLLOW Statistics and Mathematics Tutorials for more clear, practical lessons in statistics, mathematics, and data analysis. #Statistics #Correlation #PearsonCorrelation #CorrelationAnalysis #DataAnalysis #DataScience #StatisticsEducation #StatisticalModeling #QuantitativeAnalysis #ResearchMethods #RegressionAnalysis #ScatterPlot #DataAnalytics #Econometrics #PredictiveAnalytics #LearnStatistics #Statisticians #StatisticsStudents #Mathematics #StatisticsAndMathematicsTutorials #SimplifyingDataEmpoweringMinds #DataDrivenDecisionMaking https://youtube.com/@statistics_india/ https://wa.me/channel/0029Vadop5ZBfxoF9Lq8h015 https://www.instagram.com/statistics_india https://t.me/statistics_india https://www.fb.com/statistics_india https://www.threads.net/@statistics_india

### What is a Normal Distribution? A normal distribution, also known as a Gaussian distribution, is a type of continuous probability distribution for a real-valued random variable. It is characterized by its bell-shaped curve, which is symmetrical around the mean. The mean, median, and mode of a normal distribution are all equal. ### Founder and Origin The normal distribution was first introduced by Abraham de Moivre in 1733. However, it was later formalized by Carl Friedrich Gauss in the early 19th century, leading to its alternative name, the Gaussian distribution. ### When Should it Be Used? The normal distribution is used when the data is symmetrically distributed with no skew. It is often applicable in situations where the data tends to cluster around a central point, such as heights, test scores, or measurement errors. ### Explanation for an Ordinary Person To explain the normal distribution to an ordinary person, you can use the following analogy: "Imagine you're measuring the heights of a large group of people. Most people will be around average height, while a few will be much taller or shorter. If you plot the number of people against their heights, you'll get a bell-shaped curve. This is a normal distribution, where the peak represents the average height, and the spread shows how much the heights vary." ### Importance in Inferential Statistics The normal distribution is crucial in inferential statistics because many statistical methods assume that the data follows a normal distribution. This allows for the use of tools like z-scores, t-tests, and confidence intervals to make inferences about a population based on sample data. It simplifies the analysis and helps in making predictions. ### Examples 1. Heights of People: If you measure the heights of 1,000 adults, most will be around the average height of, say, 170 cm. A normal distribution will show that few people are extremely tall or short, and most are around the average. 2. Test Scores: In a large class, most students' test scores will cluster around the average score, with fewer students scoring very high or very low. This pattern will form a normal distribution. 3. Measurement Errors: When you measure something repeatedly, small errors will follow a normal distribution, with most measurements being close to the true value and fewer measurements deviating significantly. ### Why it is Important in Inferential Statistics? The normal distribution is important in inferential statistics for several reasons: 1. Central Limit Theorem: This theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed, regardless of the original distribution. This makes the normal distribution a good approximation for many real-world phenomena. 2. Standardization: It allows for data to be standardized, making it easier to compare different data sets and perform statistical tests. 3. Probability Calculation: The properties of the normal distribution make it easier to calculate probabilities and make predictions about a population. By understanding and utilizing the normal distribution, statisticians can make more accurate and reliable inferences from sample data to the broader population. https://t.me/statistics_india

✓✓ Definition of Type I Error In statistics, type I error is defined as an error that occurs when the sample results cause the rejection of the null hypothesis, in spite of the fact that it is true. In simple terms, the error of agreeing to the alternative hypothesis, when the results can be ascribed to chance. Also known as the alpha error, it leads the researcher to infer that there is a variation between two observances when they are identical. The likelihood of type I error, is equal to the level of significance, that the researcher sets for his test. Here the level of significance refers to the chances of making type I error. E.g. Suppose on the basis of data, the research team of a firm concluded that more than 50% of the total customers like the new service started by the company, which is, in fact, less than 50%. ✓✓ Definition of Type II Error When on the basis of data, the null hypothesis is accepted, when it is actually false, then this kind of error is known as Type II Error. It arises when the researcher fails to deny the false null hypothesis. It is denoted by Greek letter ‘beta (β)’ and often known as beta error. Type II error is the failure of the researcher in agreeing to an alternative hypothesis, although it is true. It validates a proposition; that ought to be refused. The researcher concludes that the two observances are identical when in fact they are not. The likelihood of making such error is analogous to the power of the test. Here, the power of test alludes to the probability of rejecting of the null hypothesis, which is false and needs to be rejected. As the sample size increases, the power of test also increases, that results in the reduction in risk of making type II error. E.g. Suppose on the basis of sample results, the research team of an organisation claims that less than 50% of the total customers like the new service started by the company, which is, in fact, greater than 50%. ✓✓ Key Differences Between Type I and Type II Error 1. Type I error is an error that takes place when the outcome is a rejection of null hypothesis which is, in fact, true. Type II error occurs when the sample results in the acceptance of null hypothesis, which is actually false. 2. Type I error or otherwise known as false positives, in essence, the positive result is equivalent to the refusal of the null hypothesis. In contrast, Type II error is also known as false negatives, i.e. negative result, leads to the acceptance of the null hypothesis. 3. When the null hypothesis is true but mistakenly rejected, it is type I error. As against this, when the null hypothesis is false but erroneously accepted, it is type II error. 4. Type I error tends to assert something that is not really present, i.e. it is a false hit. On the contrary, type II error fails in identifying something, that is present, i.e. it is a miss. 5. The probability of committing type I error is the sample as the level of significance. Conversely, the likelihood of committing type II error is same as the power of the test. #Statistics https://t.me/statistics_india

🔵 π (PI) | The Mathematical Constant 📌 Definition π (Pi) is the ratio of the circumference of a circle to its diameter. 📊 π in Statistics & Probability π appears naturally in the Normal Distribution: 📈 Normal probability density 📊 Statistical inference 🎲 Probability theory 🔬 Mathematical modelling 📐 Geometry & trigonometry ⚙️ Engineering & physics 🎯 Remember: π ≠ 22/7 exactly Therefore, its decimal expansion is non-terminating and non-repeating. #Pi #Mathematics #Statistics #Probability #NormalDistribution #MathematicalConstant #GPSC #GSS #ResearchOfficer #StatisticsIndia #CompetitiveExams

📊SYSTEMATIC RANDOM SAMPLING 🎯 Definition Systematic Random Sampling is a probability sampling method in which units are selected from an ordered population at a fixed sampling interval after choosing a random starting point. [ \boxed{k=\frac{N}{n}} ] where N= Population size n= Required sample size k= Sampling interval ⭐Advantages ✅Simple & easy to implement ✅Faster than SRS for large populations ✅Gives good population coverage ✅Requires only a single random start ✅Useful for field surveys and ordered lists ⚠️Limitations ❌Periodicity in the population can cause serious bias. ❌Requires a suitable sampling frame. ❌If the ordering has a hidden pattern, the sample may not be representative. 🎯Memory Trick: “R + K = Systematic Sampling” 🎲R = Random Start 📏K = Fixed Interval 🔥DIFFERENCE SRS: Random selection throughout🎲 Systematic: Random start + every k^{th} unit📏 📢1-Line Revision

Choose a random starting point→ select every k^{th} unit

#Sampling #SystematicSampling #SamplingTheory #ResearchOfficer

📊 SYSTEMATIC RANDOM SAMPLING | Statistics Quick Revision 🎯 Definition Systematic Random Sampling is a probability sampling method in which units are selected from an ordered population at a fixed sampling interval after choosing a random starting point. [ \boxed{k=\frac{N}{n}} ] where N = Population size n = Required sample size k = Sampling interval --- 🔢 Selection Procedure 👥 Population N ⬇️ 📐 Calculate: [ k=\frac{N}{n} ] ⬇️ 🎲 Select a random starting number: [ r\in{1,2,\ldots,k} ] ⬇️ 📌 Select: [ \boxed{r,;r+k,;r+2k,;r+3k,\ldots} ] --- 📝 Example Suppose: [ N=1000,\qquad n=100 ] Therefore, [ k=\frac{1000}{100}=\boxed{10} ] If the random starting point is: [ r=7 ] then selected units are: [ \boxed{7,;17,;27,;37,;47,\ldots,997} ] 🎯 Total selected units = 100 --- 📈 Simple Illustration Population: 1 2 3 4 5 6 7⭐ 8 9 10 11 12 13 14 15 16 17⭐18 19 20 21 22 23 24 25 26 27⭐28 29 30 ↑ Every 10th unit --- ⭐ Advantages ✅ Simple & easy to implement ✅ Faster than SRS for large populations ✅ Gives good population coverage ✅ Requires only a single random start ✅ Useful for field surveys and ordered lists ⚠️ Limitations ❌ Periodicity in the population can cause serious bias. ❌ Requires a suitable sampling frame. ❌ If the ordering has a hidden pattern, the sample may not be representative. --- 🧠 EXAM REMINDER [ \boxed{k=\frac{N}{n}} ] [ \boxed{\text{Random Start}+ \text{Fixed Interval}} ] 🎯 Memory Trick: “R + K = Systematic Sampling” 🎲 R = Random Start 📏 K = Fixed Interval 🔥 DIFFERENCE SRS: Random selection throughout 🎲 Systematic: Random start + every k^{th} unit 📏 --- 📢 1-Line Revision «Choose a random starting point → select every k^{th} unit.» #Statistics #Sampling #SystematicSampling #SamplingTheory #GPSC #GSS #ResearchOfficer #StatisticsIndia #CompetitiveExams #Statistics #Probability

📚STRATIFIED RANDOM SAMPLING 🎯 Definition When a population is divided into homogeneous subgroups (strata) and a random sample is selected from each stratum, the method is called Stratified Random Sampling. 📌 Proportional Allocation N_h = population size of stratum h n_h= sample size from stratum h N=total population n=total sample ⭐Advantages ✅Better representation of important subgroups ✅Often gives greater precision than SRS ✅Useful for heterogeneous populations ✅Enables separate estimates for each stratum ✅Can reduce sampling variance ⚠️Important Conditions 🔹Strata should ideally be internally homogeneous. 🔹Different strata should be as distinct as possible. 📍A proper sampling frame should be available for each stratum. 🧠REMINDER STRATIFIED=DIVIDE → RANDOMLY SELECT 📌Homogeneous WITHIN strata 📌Heterogeneous BETWEEN strata 🎯Memory Trick “Same inside, different outside” #Statistics #Sampling #StratifiedSampling #ResearchOfficer #GPSC #GSS #StatisticsIndia #SamplingTheory 📲 @Statistics_india

📊Simple Random Sampling(SRS) 🎯Definition Simple Random Sampling is a probability sampling method in which every unit of the population has an equal and known chance of being selected. 🔢Basic Notation N = Population Size n = Sample Size For SRS without replacement: n/N 🔄Two Types 1️⃣SRSWR — With Replacement Selected unit is returned before the next draw. 2️⃣SRSWOR — Without Replacement Selected unit is not returned. 📌Exam favourite: SRSWOR ✅Advantages ✔️Simple and unbiased selection ✔️Easy to understand ✔️Suitable when a complete sampling frame is available ✔️Sampling error can be estimated ⚠️Limitations ❌Requires a complete population list ❌Can be costly for geographically scattered populations ❌May not adequately represent small subgroups 🧠 REMEMBER SRS = Equal Chance + Random Selection + Known Probability 🎯 Population → Random Selection → Sample → Estimation #Statistics #Sampling #SimpleRandomSampling #SRS #StatisticsIndia #GPSC #GSS #ResearchOfficer #SamplingMethods 📲@statistics_india

🔆 Centre’s Fiscal Outlook Faces Geopolitical & Revenue Risks 📍 Why in News? ✅ The Centre’s fiscal position for 2026–27 may remain broadly on track, but faces risks from weak tax revenue growth, higher subsidies and West Asian geopolitical tensions. 📍 Revenue & Taxation ✅ Centre’s Gross Tax Revenue (GTR) grew only 3.7% in Q1 2026–27. ✅ PIT growth: 6.8%; GST revenue growth: contracted by 11% in Q1. ✅ Excise-duty cuts on petrol and diesel reduced revenue, while global crude prices increased subsidy pressures. ✅ Compensatory measures include Health Security and National Security (HSNS) Cess and higher import duties on selected precious metals. 📍 Fiscal Deficit & Debt ✅ Q1 fiscal deficit accounted for 18.2% of the annual budgeted magnitude. ✅ Full-year fiscal deficit is estimated at around 4.6% of GDP. ✅ Debt-to-GDP ratio: estimated at 55.8%, close to the budgeted level. 📍 Transfers & RBI Support ✅ States’ share in the divisible pool remains at 41% under the Sixteenth Finance Commission framework. ✅ RBI dividends provided significant fiscal support, with 77% of the budgeted dividends and profits already received in the first three months. 📍 Key Risks ✅ West Asian crisis: Higher crude prices can increase subsidies and worsen the fiscal position. ✅ Revenue shortfall: Weak GST and PIT growth may constrain tax collections. ✅ Higher subsidies: Annual subsidy requirements could exceed the budgeted amount. ✅ External debt & currency pressures may add to fiscal stress. 🎯 Mains Application: A credible fiscal strategy requires balancing fiscal consolidation with revenue mobilisation and targeted expenditure, while building buffers against geopolitical and commodity-price shocks. ✎ Mains Question: Geopolitical tensions can increasingly influence India’s fiscal stability through both revenue and expenditure channels. Discuss. #Economy

Econometrics - અર્થમિતિશાસ્ત્ર books

📊Chi-Square Goodness-of-Fit Test The Chi-Square Goodness-of-Fit Test is an important non-parametric test used to determine whether the observed frequencies of categorical data are consistent with a specified theoretical or expected distribution. 1.Definition The Chi-Square Goodness-of-Fit Test tests whether the difference between Observed Frequencies and Expected Frequencies is statistically significant. 2.Objectives of the Test The Goodness-of-Fit test is used to determine whether: Observed frequencies agree with theoretical frequencies. A sample follows a specified probability distribution. A claimed population distribution is reasonable. Differences between observed and expected frequencies can be attributed to chance variation. 3. Assumptions ① Data should be frequency data The test is primarily applied to counts/frequencies, not raw measurements. ② Categories should be mutually exclusive An observation should belong to only one category. ③ Observations should be independent 4. D.F. = k-1 👍🏻📲🔔

📘Gamma Distribution 🎯Definition The Gamma Distribution is a continuous probability distribution commonly used to model waiting time until a specified number of events occur X∼Gamma(α,θ)​ where: = shape parameter = scale parameter 📈Shape of Gamma Distribution The shape depends mainly on α: f(x) ↑ | ╭──╮ | ╭─╯ ╰──── | ╭─╯ | ╭─╯ |╯ +────────────────→ x Waiting time 0<α<1:: strongly right-skewed α=1: Exponential distribution α>1: generally unimodal and right-skewed Large α : distribution becomes more symmetric 🧠Memory Box Mean=αθ Variance=αθ^2 SD=θ/`α ​Mode=(α−1)θ Exponential=α=1 Gamma Function=Γ(n)=(n−1)! Rate=λ=1/θ 🔥Remember Gamma → Waiting Time → Shape + Scale Gamma⇒α,θ 👇🏼👍🏻🔔📲 https://youtube.com/@statistics_india/ https://wa.me/channel/0029Vadop5ZBfxoF9Lq8h015 https://www.instagram.com/statistics_india https://t.me/statistics_india https://www.fb.com/statistics_india https://www.threads.net/@statistics_india

Probability = Area of Normal Distribution 🔔