Stats1_Week2_Descriptive_Stats

Stats 1 Week 2: Descriptive Statistics & Visualization

0. Prerequisites

NOTE

What you need to know:

• Percentages: Part/Total * 100.
• Fractions: 1/4 = 0.25 = 25%.
• Basic Graphs: Reading X and Y axes.

Quick Refresher

• Frequency ($f$): How many times something happens.
• Relative Frequency ($rf$): Proportion of total. $rf=f/N$.
• Cumulative Frequency ($cf$): Running total.

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1. Core Concepts

1.1 Frequency Distributions

• Categorical Data:
  • Frequency Table: Lists categories and counts.
  • Relative Frequency: Useful for comparing groups of different sizes.
• Numerical Data:
  • Grouped Frequency: Bins (e.g., 0-10, 10-20).
  • Class Width: Upper Limit - Lower Limit.

1.2 Graphical Representations

1. Bar Chart: For Categorical data. Bars have gaps. Height = Frequency.
2. Pie Chart: For Categorical data (Parts of a Whole).
   • Angle = $rf\times 360^{\circ }$.
3. Pareto Chart: Bar chart sorted by frequency (High to Low) + Cumulative line. Used to find “Vital Few”.
4. Histogram: For Numerical data. Bars touch. Area represents frequency.
5. Stem & Leaf Plot: Preserves actual data values.
   • Example: 3 | 2 5 means 32, 35.

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2. Pattern Analysis & Goated Solutions

Pattern 1: Reading Pie Charts (Aggregate Distribution)

Context: “Pie chart shows percentages. Total marks 500. Find total marks in Physics + Math.”

TIP

Mental Algorithm:

1. Sum Percentages: Add the % for the required categories.
2. Calculate Value: $\frac{\text{Sum \%}}{100}\times \text{Total Value}$.

Example (Detailed Solution)

Problem: Total marks 500. Physics (25%), Math (30%), Bio (20%). Find aggregate marks. Solution:

1. Sum %: $25+30+20=75\%$.
2. Calculate:
   • $75\%$ of 500.
   • $0.75\times 500=375$. Answer: 375 marks.

Pattern 2: Stem and Leaf Plot Analysis

Context: “Find the Median from Stem and Leaf plot.” Data: 2 | 1 5 3 | 0 2 2 6 4 | 1

TIP

Mental Algorithm:

1. List Data: Decode the plot into a sorted list.
2. Count ($N$): Total observations.
3. Find Middle:
   • If $N$ is Odd: $(N+1)/2$-th term.
   • If $N$ is Even: Average of $N/2$ and $(N/2)+1$-th terms.

Example (Detailed Solution)

Problem: Find median of above plot. Solution:

1. List: 21, 25, 30, 32, 32, 36, 41.
2. Count: $N=7$ (Odd).
3. Position: $(7+1)/2=4$-th term.
4. Identify: 1st(21), 2nd(25), 3rd(30), 4th(32). Answer: 32.

Pattern 3: Combined Relative Frequency

Context: “Find combined relative frequency of Academy A, B, and D.”

TIP

Mental Algorithm:

1. Sum Frequencies: Add counts for A, B, D.
2. Total Count: Find sum of ALL categories.
3. Divide: $\frac{\text{Sum (A+B+D)}}{\text{Total}}$.

Example (Detailed Solution)

Problem: A(10), B(20), C(30), D(40). Find combined RF of A, B, D. Solution:

1. Sum A+B+D: $10+20+40=70$.
2. Total: $10+20+30+40=100$.
3. Divide: $70/100=0.7$. Answer: 0.7.

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3. Practice Exercises

1. Pie Chart: If 20% is “Red” and Total is 200, how many Red?
   • Hint: $0.2\times 200=40$.
2. Histogram: Can you find the exact mode from a histogram?
   • Hint: No, only the modal class (interval).
3. Bar vs Histogram: Which one for “Height of students”?
   • Hint: Height is continuous numerical $\to$ Histogram.