Descriptive Statistics

2. Population vs Sample

2.1 🌍 What is Population?
1. A population is the entire group we want to study. Examples: - All students in a school - All customers of a company - All people living in a city - All products sold by a shop
2.2 👥 What is Sample?
1. A sample is a smaller group selected from the population. We study the sample to understand the larger population. If a city has 1 million people, it may be difficult to ask everyone about their income. So, we may randomly select 1,000 people and study them. Here: Population = 1 million people Sample = 1,000 selected people
2.3 🤔 Why Do We Use Samples?
1. We use samples because studying the entire population can be: - Too expensive 💰 - Too time-consuming ⏰ - Difficult to manage 📋 - Sometimes impossible 😵 So, we study a smaller group and use it to make conclusions about the larger group.
2.4 🎯 Representative Sample
1. A sample should fairly represent the population, meaning it should give a balanced picture of the whole group. If we want to know the opinion of all students in a school, we should not ask only Grade 10 students. We should include students from different grades. 📌 A good sample gives a fair picture of the whole population.

2.5 ⚠️ Sampling Bias
1. Sampling bias happens when the sample does not properly represent the population. If we ask only cricket fans about their favorite sport, most may say cricket. But that does not mean everyone likes cricket the most. 📌 A biased sample can lead to wrong conclusions.

3. Statistics

3.1 📊 What is Statistics?
1. Statistics is the science of: ✅ Collecting data ✅ Organizing data ✅ Analyzing data ✅ Understanding data ✅ Presenting data clearly In simple words: Statistics helps us make sense of data.
3.2 📋 Types of Statistics
1. There are two main types of statistics: - Descriptive Statistics - Inferential Statistics

4. Central Tendency

4.1 🎯 What is Central Tendency?
1. Central tendency tells us the center or typical value of data. It helps answer: What value represents this data best? The three most common measures are: - Mean - Median - Mode
4.2 ➗ Mean
1. Mean is the arithmetic average of numbers. Formula: Mean = Sum of all values / Number of values For data: [2, 1, 2, 5, 100], mean = (2 + 1 + 2 + 5 + 100) / 5 = 22 ⚠️ Mean can be affected by very large or very small values called outliers. 📌 In this example, 100 makes the mean much larger than most values.
4.3 📍 Median
1. Median is the middle value after sorting the data. Steps: 1. Sort the data 2. Find the middle value Example: Original data: [2, 1, 2, 5, 100] Sorted data: [1, 2, 2, 5, 100] Middle value: 2 So, the median is 2. ✅ Median is useful when data has outliers. 📌 Here, median 2 represents the data better than mean 22.

4.4 🔁 Mode
1. Mode is the value that appears most often. Example: For data: [2, 1, 2, 5, 100] The value 2 appears most often, so the mode is 2. data = ["Red", "Blue", "Red", "Green", "Red"] Mode is Red because it appears most often. Mode works for both numerical and categorical data.

5. Percentiles and Quartiles

5.1 📍 What are Percentiles and Quartiles?
1. Percentiles and quartiles tell us the position of a value compared to the rest of the data. They help answer: Is this value high, low, or average compared to others?
5.2 🎓 Simple Example
1. Suppose a student scores 60 / 100. Is this good or bad? We cannot say clearly unless we know how other students scored. - If most students scored below 60, then 60 is good. ✅ - If most students scored above 60, then 60 may not be good. ⚠️ So, position matters.
5.3 📊 Percentiles
1. Percentiles divide data into 100 equal parts. The p-th percentile means: p% of the data is below that value. If a student is in the 90th percentile, it means: The student scored better than 90% of the students.
5.4 🧱 Quartiles
1. Quartiles divide data into 4 equal parts. The three main quartile values are Q1, Q2, and Q3.

Interquartile Range: IQR
1. IQR shows the spread of the middle 50% of the data. Formula: IQR = Q3 - Q1 IQR is useful because it is not strongly affected by outliers. Detecting Outliers Using IQR Values are often considered outliers if they are: Below Q1 - 1.5 × IQR (lower limit) or Above Q3 + 1.5 × IQR (upper limit) 📌 IQR helps us find values that are unusually small or unusually large.

6. Dispersion

6.1 📏 What is Dispersion?
1. Dispersion tells us how spread out the data is. It helps answer: Are the values close together or far apart? Example: Dataset A: [38, 40, 42] Dataset B: [5, 20, 95] Both datasets have mean 40. But Dataset B is much more spread out. That spread is called dispersion.
6.2 Common Measures of Dispersion
1. The common measures are: - Range - Variance - Standard Deviation
6.3 📐 Range
1. Range is the difference between the maximum and minimum value. Formula: Range = Max - MinExample: For data: [2, 1, 2, 5, 100] Range: 100 - 1 = 99 So, the range is 99. 📌 Range gives a quick idea of spread, but it is highly affected by outliers.
6.4 📊 Variance
1. Variance tells us how spread out values are from the mean. Simple meaning: - High variance ➜ Values are spread far from the mean - Low variance ➜ Values are close to the mean Formula: Variance = Σ(Xi - X̄)² / N

6.5 📉 Standard Deviation
1. Standard deviation is the square root of variance. Formula: Standard Deviation = √Variance Standard deviation is easier to understand than variance because it is in the same unit as the original data. 📌 Standard deviation tells us how much values usually differ from the average.

7. Example: Dispersion Calculation

7.1 Given Data
1. For data points [5, 20, 95]: Mean: (5 + 20 + 95) / 3 = 120 / 3 = 40 Minimum: 5 Maximum: 95 Range: 95 - 5 = 90

8. Covariance and Correlation

8.1 🔗 Why Do We Study Relationships Between Variables?
1. In Data Science, we often want to know how two variables are related. Examples: - Does more study time lead to higher marks? - Does more advertising increase sales? - Does temperature affect ice cream sales? - Does income affect spending? To study relationships, we use: Covariance and Correlation
8.2 🔄 Covariance
1. Covariance tells us whether two variables move together. Formula: Cov(X, Y) = Σ((Xi - X̄)(Yi - Ȳ)) / N

8.3 📈 Correlation
1. Covariance tells us the direction of a relationship, but not how strong it is. Correlation tells us both: ✅ Direction of relationship ✅ Strength of relationship Correlation always lies between -1 and +1. Formula: Correlation = Cov(X, Y) / (Std(X) × Std(Y))

9. Example: Covariance and Correlation Calculation

9.1 Given Data
1. X = [5, 20, 95] Y = [10, 30, 50] Mean of X: X̄ = (5 + 20 + 95) / 3 = 40 Mean of Y: Ȳ = (10 + 30 + 50) / 3 = 30

Calculations
1. Var(X) = (1225 + 400 + 3025) / 3 = 4650 / 3 = 1550 Var(Y) = (400 + 0 + 400) / 3 = 800 / 3 = 266.67 Cov(X, Y) = (700 + 0 + 1100) / 3 = 1800 / 3 = 600 Std(X) = √1550 = 39.37 Std(Y) = √266.67 = 16.33Correlation = 600 / (39.37 × 16.33) = 600 / 642.86 = 0.93 📌 Interpretation: Correlation is 0.93, which means X and Y have a strong positive relationship.

10. Spearman Rank Correlation

10.1 🏆 What is Spearman Rank Correlation?
1. Spearman Rank Correlation is used when data has rank or order. It is useful for ordinal data. Examples of ordinal data: - T-shirt size: S, M, L, XL - Rating: Poor, Average, Good, Excellent - Rank in class: 1st, 2nd, 3rd - Satisfaction level: Low, Medium, High T-shirt sizes have order: S < M < L < XL. So, we can rank as: S = 1, M = 2, L = 3, XL = 4Formula: ρ = 1 - (6 × Σd²) / (n × (n² - 1))
10.2 Calculation Example
1. T-shirt Size = ['M', 'S', 'L', 'XL'] Customer Satisfaction = [3, 4, 2, 5]

Ranking Table
1. The table above shows each T-shirt size and satisfaction score converted into ranks, along with the rank difference (d) and squared difference (d²) needed for the formula. Calculationn = 4Σd² = 0 + 4 + 4 + 0 = 8n(n² - 1) = 4(4² - 1) = 4(16 - 1) = 60ρ = 1 - (6 x 8) / 60 = 1 - 48/60 = 1 - 0.8 = 0.2 📌 Interpretation: Spearman correlation is 0.2, which shows a weak positive relationship between T-shirt size and customer satisfaction.

11. Probability

11.1 🎲 What is Probability?
1. Probability tells us how likely something is to happen. It is a number between 0 and 1. Interpretation: 0 ➜ Impossible 0.5 ➜ 50% chance 1 ➜ Certain ☔ Simple Example If a weather app says: Probability of rain = 0.8 It means there is an 80% chance of rain, so carrying an umbrella is a smart choice.
11.2 Probability Formula
1. P(A) = Number of favorable outcomes / Total number of outcomesExample: If a coin is flipped, possible outcomes are: Head, Tail Probability of getting Head: P(Head) = 1 / 2 = 0.5 So, there is a 50% chance of getting Head.

11.3 Basic Probability Rules
1. Rule 1: Event and Not Event P(A) + P(not A) = 1 Example: If probability of rain is 0.8: P(no rain) = 1 - 0.8 = 0.2Rule 2: A or B P(A or B) = P(A) + P(B) - P(A and B) This is used when we want either event A or event B to happen.