STAT 17: Statistical Methods for Business and Economics

Prof. Marcela Alfaro Cordoba

Statistics - UCSC

28 Apr 2026

The Netflix Data Challenge 📈

After solving the recommendation mystery, Sarah now faces a new challenge…

She has 50,000 user survey responses with:

Weekly watch times
Satisfaction ratings (1-5 scale)
Preferred genres
Age demographics

The Problem: Raw numbers everywhere! How can she make sense of it all? 🤔

Today’s mission: Learn to visualize data and calculate measures of center!

What We’ll Accomplish Today

By the end of this lecture, you will be able to:

✅ Create and interpret histograms, box plots, and charts using Google Sheets
✅ Calculate and interpret measures of central tendency (mean, median, mode)
✅ Use sigma notation and calculate arithmetic mean properly
✅ Choose the right measure of central tendency for different scenarios

Why Visualize Data?

Raw Numbers 😵

12, 15, 8, 22, 18, 9, 14, 
16, 20, 11, 13, 17, 19, 10, 
15, 21, 14, 16, 18, 12, 15, 
19, 13, 17, 20, 25, 8, 14, 
16, 22, 11...

Hard to see patterns!

Visualized Data 😊

Patterns emerge immediately!

Key insight: Human brains process visual information 60,000x faster than text!

Types of Data Displays

Three Main Categories

Histograms - Show distribution of continuous numerical data
Bar Charts - Compare categories
Pie Charts - Show proportions of a whole

Let’s explore each one with Sarah’s Netflix data! 🎬

Histograms: Showing Distributions 📊

What is a Histogram?: A bar chart showing the frequency distribution of continuous numerical data by grouping values into bins (ranges)

Key Features:

Bars touch each other (no gaps!)
X-axis: ranges of values (bins)
Y-axis: frequency or count
Shows the shape of the data

Histogram Example

Sarah’s Data: Weekly watch time (hours) for 100 users

What do you notice? Most users watch 10-15 hours weekly!

Creating Histograms in Google Sheets 💻

Steps:

Enter your data in a column (e.g., A1:A100)
Select the data range
Click Insert → Chart
In Chart Editor, choose Chart type → Histogram
Customize:
- Add chart title
- Label axes
- Adjust bin size

Pro tip: Experiment with different bin sizes - too few bins hide patterns, too many create noise!

Bar Charts vs Histograms

Bar Charts

For categorical data
Bars have gaps
Order doesn’t matter (usually)
Compare groups

Example: Number of users per genre preference

Histograms

For continuous data
Bars touch
Order matters
Show distribution shape

Example: Distribution of watch times

Bar Chart Example

Sarah’s Genre Preferences:

Creating in Google Sheets: Insert → Chart → Column chart (or Bar chart)

Pie Charts: Part of a Whole 🥧

When to Use:

Show proportions or percentages
Limited number of categories (3-6 ideal)
When parts add up to 100%

When NOT to Use:

Comparing many categories (use bar chart)
Showing trends over time (use line chart)
When precise values matter

Pie Chart Example

Subscription Type Distribution:

Google Sheets: Insert → Chart → Pie chart

Box Plots: The Five-Number Summary 📦

Box Plot shows the distribution through five key values:

Minimum - smallest value
Q1 (First Quartile) - 25th percentile
Median - 50th percentile (middle value)
Q3 (Third Quartile) - 75th percentile
Maximum - largest value

Bonus: Easily spots outliers! 🎯

Box Plot Anatomy

Red dots = Outliers (values far from the rest)

Creating Box Plots in Google Sheets

Unfortunately, Google Sheets doesn’t have a built-in box plot option! 😔

Solutions:

Create manually using candlestick charts (advanced)
Use add-ons like “Box and Whisker Chart”
Calculate five-number summary and describe it

For now: Focus on calculating the five numbers using Google Sheets functions!

Measures of Central Tendency 🎯

Sarah asks: “What’s a typical Netflix user’s weekly watch time?”

She has 50,000 different numbers. Which single number best represents them all?

Three measures of “center”:

Mean (average)
Median (middle value)
Mode (most frequent)

The Mean (Average) 📊

Definition: The sum of all values divided by the number of values

Why use it?:

Uses ALL data points
Algebraically useful for further calculations
Most commonly reported measure

Caution: Sensitive to outliers! 🚨

Mean Formulas and Notation

Sample Mean: \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} = \frac{x_1 + x_2 + ... + x_n}{n}\)

Population Mean: \(\mu = \frac{\sum_{i=1}^{N} x_i}{N}\)

Understanding Sigma Notation (Σ):

Σ (capital Greek letter sigma) means “sum of”
i=1 is the starting index
n or N is the ending index
x_i represents each value in the dataset

Sigma Notation Example

Calculate: \(\sum_{i=1}^{5} x_i\) where x = {10, 15, 12, 18, 20}

Expanding:

\(\sum_{i=1}^{5} x_i = x_1 + x_2 + x_3 + x_4 + x_5\)

\(= 10 + 15 + 12 + 18 + 20 = 75\)

Mean: \(\bar{x} = \frac{75}{5} = 15\) hours

Calculating Mean in Google Sheets

Example data in cells A1:A5: 10, 15, 12, 18, 20

Formula: =AVERAGE(A1:A5)

Result: 15

Alternative (manual calculation):

=SUM(A1:A5)/COUNT(A1:A5)

Also gives 15!

Pro tip: Use AVERAGE for simplicity, but know the underlying calculation!

The Median 🎯

Definition: The middle value when data is arranged in order

How to find it:

Order the data from smallest to largest
If n is odd: The median is the middle value
If n is even: The median is the average of the two middle values

Median Examples

Odd n (n=5):

Data: 10, 12, 15, 18, 20

Median = 15 hours

(The 3rd value)

Even n (n=6):

Data: 10, 12, 15, 18, 20, 22

Median = (15 + 18)/2 = 16.5 hours

(Average of 3rd and 4th values)

Position formula: If ordered, median is at position (n+1)/2

Calculating Median in Google Sheets

Example data in A1:A6: 20, 10, 15, 12, 22, 18

Formula: =MEDIAN(A1:A6)

Result: 16.5

Google Sheets automatically:

Sorts the data
Finds the middle value(s)
Averages if n is even

Key advantage: You don’t need to manually sort! 🎉

The Mode 🔢

Definition: The value that appears most frequently in the dataset

Types:

Unimodal: One mode (most common)
Bimodal: Two modes
Multimodal: More than two modes
No mode: All values appear equally (or all unique)

Mode Examples

Unimodal:

12, 15, 15, 15, 18, 20

Mode = 15

(appears 3 times)

Bimodal:

12, 12, 15, 18, 18, 20

Modes = 12 and 18

(both appear twice)

No Mode:

10, 12, 15, 18, 20

No mode

(all values unique)

Best for:

Categorical data!

Example: Most popular genre, common subscription type

Calculating Mode in Google Sheets

Example data in A1:A7: 15, 12, 15, 18, 15, 20, 12

Formula: =MODE(A1:A7)
or
=MODE.SNGL(A1:A7)

Result: 15

For multiple modes: =MODE.MULT(A1:A7) returns array of modes

Limitation: Only works with numerical data. For categorical data, use pivot tables or COUNTIF!

Mean vs Median: A Critical Comparison

Sarah’s Dilemma: Weekly watch times (hours) from 7 users

Data: 5, 8, 10, 12, 15, 18, 95

Mean: (5+8+10+12+15+18+95)/7 = 23.3 hours

Median: 5, 8, 10, 12, 15, 18, 95 = 12 hours

Question: Which better represents a “typical” user? 🤔

When Mean ≠ Median

The 95-hour user is an OUTLIER - someone who binges excessively!

Impact:

Mean is “pulled” toward the outlier (23.3 hrs - inflated!)
Median is resistant to outliers (12 hrs - more typical!)

Sarah’s decision: Report median to executives when data has outliers!

Which Measure to Use? 🤔

Use MEAN when:

Data is symmetric (no extreme values)
You want to use all data
Doing further calculations
Working with interval/ratio data

Example: Average test scores in a class

Use MEDIAN when:

Data has outliers or is skewed
You want a resistant measure
Working with ordinal data
Describing “typical” value

Example: Household income, home prices

Which Measure to Use? (continued)

Use MODE when:

Working with categorical data
Want the most typical/popular value
Interested in what occurs most often

Example: Most popular shoe size, favorite ice cream flavor

Pro tip: Often report multiple measures for complete picture!

Example: “Mean watch time is 23 hours, but median is 12 hours, suggesting some extreme binge-watchers.”

Google Sheets: Hands-On Summary 💻

Essential Functions:

Function	Purpose	Example
`=AVERAGE(range)`	Calculate mean	`=AVERAGE(A1:A100)`
`=MEDIAN(range)`	Find median	`=MEDIAN(A1:A100)`
`=MODE(range)`	Find mode	`=MODE(A1:A100)`
`=SUM(range)`	Sum all values	`=SUM(A1:A100)`
`=COUNT(range)`	Count values	`=COUNT(A1:A100)`
`=MIN(range)`	Minimum value	`=MIN(A1:A100)`
`=MAX(range)`	Maximum value	`=MAX(A1:A100)`

Creating Charts in Google Sheets

Steps:

Select your data range (including headers if you have them)
Click Insert → Chart
Choose chart type in Chart Editor:
- Histogram for distributions
- Column/Bar chart for categories
- Pie chart for proportions
Customize:
- Add chart title and axis labels
- Adjust colors and styling
- Modify legend position

Sarah’s Success Story! 🎉

What Sarah Discovered:

Histogram revealed most users watch 10-15 hours/week (clear peak)
Mean = 18 hours (pulled up by binge-watchers)
Median = 12 hours (better represents typical user)
Mode genre = Drama (informed content acquisition strategy)
Box plot identified super-users (outliers) for targeted retention campaigns

The Impact:

✅ 15% increase in user engagement
✅ Better content recommendations
✅ Informed marketing strategies

Your Statistical Toolkit 🧰

Visualization Skills:

Histograms show distributions of continuous data
Bar charts compare categories
Pie charts show proportions
Box plots reveal five-number summary and outliers

Central Tendency Mastery:

Mean = arithmetic average (Σx/n) - use for symmetric data
Median = middle value - resistant to outliers
Mode = most frequent - best for categorical data

Google Sheets Proficiency:

Use built-in functions: AVERAGE, MEDIAN, MODE
Create insightful visualizations with Insert → Chart
Understand when to use each measure

Looking Ahead

Next Class: Measures of Spread & Distribution Shape

Range, variance, standard deviation
Understanding skewness
Interpreting data variability

Questions? These concepts are fundamental - make sure you’re comfortable! 🤝

Quick Knowledge Check ✅

Rate your confidence (1-5) on Ed Discussion:

Creating and interpreting histograms, bar charts, pie charts ⭐⭐⭐⭐⭐
Understanding when to use each visualization type ⭐⭐⭐⭐⭐
Calculating and interpreting mean, median, mode ⭐⭐⭐⭐⭐
Using sigma notation correctly ⭐⭐⭐⭐⭐
Choosing appropriate measures for different data scenarios ⭐⭐⭐⭐⭐

If you rated anything 3 or below, please visit office hours! 🤝

Thank you! 📊✨

Questions? Office hours information on Canvas.

Next up: Measures of Spread & Distribution Shape!