Frequency Tables and Data Visualization

📹 Video Overview

🎯 What We're Learning Today

Main Topics:

1. How to organize messy data into frequency tables

2. How to visualize data with graphs

3. Different methods for different variable types

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🗂️ What is a Frequency Table?

Imagine: You have 50 test scores written randomly on paper. How do you make sense of it?

Frequency Table = A way to organize data so you can actually understand it

Why Bother?

✓ Takes huge messy data → makes it clean and organized

✓ Helps you see patterns ("Most students scored 80-90")

✓ Foundation for all future statistical analysis

✓ Helps choose the right statistical methods

💡 Memory hack: Think of it like organizing your closet - instead of clothes everywhere, you sort by type (shirts, pants, shoes) and count how many of each you have.

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🔀 The Golden Rule

Remember: Different variables → Different tables → Different graphs!

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📊 Method 1: Qualitative Variables

Example: "Which social network would you keep?"

Raw data: 73 students answered: Instagram, Facebook, Instagram, Twitter, Instagram...

The Frequency Table:

Social Network (x)	f(x) - Frequency	p(x) - Relative Frequency
Instagram	43	59%
Facebook	16	22%
Twitter	4	5%
TikTok	2	3%
LinkedIn	1	1%
None	7	10%
Total	73	100%

📐 The Key Formula:

$p(x) = \frac{f(x)}{n}$

Where:

• p(x) = relative frequency (percentage)

• f(x) = absolute frequency (count)

• n = total number of observations

Example calculation:

• Instagram: p(x) = 43/73 = 0.589 = 59%

• Facebook: p(x) = 16/73 = 0.219 = 22%

💡 Memory hack: p(x) is just "what PORTION out of everyone" - that's why it's between 0% and 100%!

📈 Graphs for Qualitative Variables:

Two options:

1. Bar Chart - Each category gets its own bar

2. Pie Chart - Shows proportions of a whole

When to use which?

• Bar chart: When comparing categories (which is bigger?)

• Pie chart: When showing parts of a whole (Instagram = 59% of the pie)

⚠️ Important: For qualitative variables, NO cumulative frequency! (Can't say "up to Facebook" - what does that even mean?)

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📊 Method 2: Discrete Quantitative Variables

Example: "Number of people in 10 families"

Raw data: 2, 2, 6, 5, 3, 5, 5, 4, 3, 2

Step 1: Count the frequencies

People in Family (x)	f(x) - Frequency	p(x) - Relative Frequency	F(x) - Cumulative Frequency	F(x)/n - Relative Cumulative
2	3	30%	3	30%
3	2	20%	5	50%
4	1	10%	6	60%
5	3	30%	9	90%
6	1	10%	10	100%
Total	10	100%	-	-

🔑 New Concept: Cumulative Frequency F(x)

What is F(x)? = "How many observations are up to and including x?"

Calculation:

• F(2) = 3 (three families have 2 people)

• F(3) = 3 + 2 = 5 (five families have up to 3 people)

• F(4) = 3 + 2 + 1 = 6 (six families have up to 4 people)

• And so on...

💡 Memory hack: F(x) is like climbing stairs - each step you ADD the previous steps!

Practice Questions:

Q1: How many families have at most 3 people?

• Answer: F(3) = 5 families

Q2: How many families have at least 3 people?

• Answer: Total - F(2) = 10 - 3 = 7 families

• (Or: everyone EXCEPT those with only 2)

📈 Graphs for Discrete Quantitative:

Two main options:

1. Bar Chart - Similar to qualitative, but order matters!

2. Frequency Polygon - Connect the tops of bars with lines

Frequency Polygon shows: The general trend in the data (is it increasing? decreasing? where's the peak?)

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📊 Method 3: Continuous Quantitative Variables

🎯 Why Group into Classes?

Problem: You have 72 students' heights: 167.3 cm, 172.8 cm, 169.5 cm...

If you list every single height, your table would be HUGE and useless!

Solution: Group into classes (ranges)

💡 Memory hack: Like organizing books by "100-200 pages", "200-300 pages" instead of listing every exact page count.

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Example 1: Student Heights (Uniform Class Width)

Height (x)	f(x)	Class Width (l)	Density (d)	p(x)	F(x)	F(x)/n
150-155	2	5	0.4	2.8%	2	2.8%
155-160	3	5	0.6	4.2%	5	6.9%
160-165	9	5	1.8	12.5%	14	19.4%
165-170	7	5	1.4	9.7%	21	29.2%
170-175	13	5	2.6	18.1%	34	47.2%
175-180	16	5	3.2	22.2%	50	69.4%
180-185	13	5	2.6	18.1%	63	87.5%
185-190	6	5	1.2	8.3%	69	95.8%
190-195	3	5	0.6	4.2%	72	100%
Total	72	-	-	100%	-	-

📐 Key Formulas for Continuous Variables:

1. Class Width (l):

$l = l_1 - l_0$

Where:

• l₁ = upper limit of class

• l₀ = lower limit of class

Example: Class 170-175

• l = 175 - 170 = 5 cm

💡 Memory hack: Class width = how WIDE is the range?

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2. Density (d):

$d = \frac{f(x)}{l}$

What is density? = Frequency per unit width

Why do we need it? Because when class widths are different, you can't compare frequencies directly!

Example: Class 170-175

• d = 13/5 = 2.6 students per cm

💡 Memory hack: Density = how PACKED/CROWDED is the class? Think of population density in cities!

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3. Percentage Density (d%):

$d\% = \frac{p(x)}{l}$

Example: Class 170-175

• d% = 18.1%/5 = 3.62% per cm

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🎨 Histogram - The Main Graph for Continuous Variables

CRITICAL RULE: When class widths are DIFFERENT, you MUST use density (d) for the y-axis!

Why? Look at this example:

Example 2: Test Scores (UNEVEN Class Width)

Scores (x)	f(x)	l	d	p(x)	d%
40-60	5	20	0.25	12.5%	0.625%
60-70	5	10	0.5	12.5%	1.25%
70-75	10	5	2	25%	5%
75-85	10	10	1	25%	2.5%
85-100	15	15	1	25%	1.67%
Total	40	-	-	100%	-

What happens if we use f(x) instead of d?

❌ WRONG: 85-100 would look like the tallest bar (15 students)

✓ CORRECT: 70-75 is actually the most DENSE (2 students per point)

The area of each rectangle in the histogram = the frequency!

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📊 Histogram vs Frequency Polygon

Feature	Histogram	Frequency Polygon
Shape	Rectangles (bars)	Connected line
Shows	Exact frequencies by class	Overall trend/pattern
Use when	Want to see distribution	Want to see shape of data

Frequency Polygon: Connect the midpoints of the top of each histogram bar

💡 Memory hack: Polygon = the "skyline" of your histogram!

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📈 Cumulative Frequency Graph

What is it? A line graph showing F(x) - how many observations are up to x

Key feature: Always goes UP (never down) and ends at 100%

Use case: "What percentage of students scored below 80?"

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🎯 Quick Decision Tree

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📝 Complete Example Walkthrough

Problem: Manhattan temperatures over 24 days in October

Temperature (°C)	f(x)	l	d	p(x)	d%
5-8	4	3	1.33	16.7%	5.6%
8-11	8	3	2.67	33.3%	11.1%
11-14	6	3	2	25%	8.3%
14-17	5	3	1.67	20.8%	6.9%
17-20	1	3	0.33	4.2%	1.4%
Total	24	-	-	100%	-

Step-by-step solution:

Step 1: Calculate class width

• All classes: l = 3°C (uniform!)

Step 2: Calculate p(x)

• 5-8: p(x) = 4/24 = 16.7%

• 8-11: p(x) = 8/24 = 33.3%

• etc.

Step 3: Calculate density

• 5-8: d = 4/3 = 1.33

• 8-11: d = 8/3 = 2.67

• etc.

Step 4: Calculate d%

• 5-8: d% = 16.7%/3 = 5.6%

• 8-11: d% = 33.3%/3 = 11.1%

• etc.

Step 5: Draw histogram

• Since widths are equal, can use f(x) OR d for y-axis

• Highest bar: 8-11°C (8 days)

Step 6: Draw frequency polygon

• Connect midpoints: 6.5°C, 9.5°C, 12.5°C, 15.5°C, 18.5°C

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🎓 Key Formulas Summary

Formula	What it calculates	When to use
p(x) = f(x)/n	Relative frequency	Any variable type
l = l₁ - l₀	Class width	Continuous variables
d = f(x)/l	Density	Continuous with classes
d% = p(x)/l	Percentage density	Continuous with classes
F(x) = Σf(x)	Cumulative frequency	Quantitative variables only

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💡 Memory Hacks Summary

1. Frequency table = organized closet - everything sorted and counted

2. p(x) = PORTION of the whole - always 0-100%

3. F(x) = climbing STAIRS - keep adding as you go up

4. Density = how PACKED/CROWDED - like population density

5. At most = ≤ (use F(x) directly)

6. At least = ≥ (use Total - F(x-1))

7. Different widths = MUST use density! - Can't compare different-sized boxes

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⚠️ Common Mistakes to Avoid

❌ Using cumulative frequency for qualitative variables

❌ Forgetting to use density when class widths differ

❌ Mixing up "at most" and "at least"

❌ Drawing histogram with f(x) when widths are unequal

❌ Forgetting that total p(x) must equal 100%

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🎯 Quick Reference Chart

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🏆 Pro Exam Tips

1. First question: What type of variable? (This determines EVERYTHING)

2. Check class widths: Equal or not? (Affects histogram)

3. Density is your friend: When in doubt with continuous variables, calculate it

4. Double-check totals: f(x) should sum to n, p(x) should sum to 100%

5. Cumulative = running total: Keep adding!

6. Area = frequency: In histograms, the area of bars represents frequency

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Remember: The type of variable is the boss - it tells you what table and graph to use!