Summarizing Data with Tables

Why Summarize Data in Tables?

Raw data is often overwhelming. Consider a dataset with 10,000 observations—you can’t understand it by staring at 10,000 numbers! Tables help us:

Organize data into manageable form
Identify patterns and common values
Communicate findings clearly
Prepare for visualization and further analysis

Frequency Tables for Categorical Data

A frequency table shows how often each category or value occurs.

Basic Frequency Table

Favorite Color Survey

A survey asked 50 people their favorite color. Raw data (abbreviated):

Blue, Red, Green, Blue, Blue, Red, Yellow, Green, Blue, …

Frequency Table:

Color	Frequency
Blue	18
Red	14
Green	10
Yellow	5
Purple	3
Total	50

Now we can immediately see that Blue is most popular, followed by Red.

Adding Relative Frequency

Relative frequency expresses each count as a proportion or percentage of the total.

Relative Frequency

$\text{Relative Frequency} = \frac{\text{Frequency}}{\text{Total Count}}$

Adding Relative Frequency

Color	Frequency	Relative Frequency	Percentage
Blue	18	$18/50 = 0.36$	36%
Red	14	$14/50 = 0.28$	28%
Green	10	$10/50 = 0.20$	20%
Yellow	5	$5/50 = 0.10$	10%
Purple	3	$3/50 = 0.06$	6%
Total	50	1.00	100%

Interpretation: 36% of respondents prefer blue—more than a third!

Frequency Tables for Numerical Data

For numerical data, we often group values into classes (intervals or bins).

Creating Classes

Guidelines for choosing classes:

Use 5-15 classes (fewer for small datasets)
Make all classes the same width
Make sure classes don’t overlap
Include all data values

Class Width

$\text{Class Width} = \frac{\text{Maximum} - \text{Minimum}}{\text{Number of Classes}}$

Round up to a convenient number.

Exam Scores Frequency Table

Data: 25 exam scores ranging from 52 to 98.

Step 1: Choose number of classes: 5 classes

Step 2: Calculate width: $(98 - 52) / 5 = 9.2 \rightarrow$ round to 10

Step 3: Create classes starting at 50:

Class	Frequency	Relative Frequency
50-59	2	0.08
60-69	4	0.16
70-79	8	0.32
80-89	7	0.28
90-99	4	0.16
Total	25	1.00

Interpretation: Most students (32%) scored in the 70s, with a fairly symmetric distribution.

Class Boundaries and Midpoints

Class boundaries eliminate gaps between classes. For the class 50-59:

Lower boundary: 49.5
Upper boundary: 59.5

Class midpoint is the center of each class:

Class Midpoint

$\text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$

Class Midpoints

Class	Midpoint
50-59	$(50 + 59)/2 = 54.5$
60-69	$(60 + 69)/2 = 64.5$
70-79	$(70 + 79)/2 = 74.5$
80-89	$(80 + 89)/2 = 84.5$
90-99	$(90 + 99)/2 = 94.5$

Midpoints are useful for estimating the mean from grouped data.

Cumulative Frequency Tables

A cumulative frequency table shows the running total of frequencies.

Cumulative Frequency

Shows how many observations fall at or below each class.

Cumulative Frequency Table

Class	Frequency	Cumulative Frequency
50-59	2	2
60-69	4	2 + 4 = 6
70-79	8	6 + 8 = 14
80-89	7	14 + 7 = 21
90-99	4	21 + 4 = 25

Interpretation:

6 students scored 69 or below
14 students scored 79 or below
21 students scored 89 or below

Cumulative Relative Frequency

Express cumulative frequencies as proportions.

Complete Frequency Table

Class	Freq	Rel. Freq	Cum. Freq	Cum. Rel. Freq
50-59	2	0.08	2	0.08
60-69	4	0.16	6	0.24
70-79	8	0.32	14	0.56
80-89	7	0.28	21	0.84
90-99	4	0.16	25	1.00

Interpretation:

24% of students scored 69 or below (bottom quarter)
56% scored 79 or below (just over half)
84% scored 89 or below

Two-Way Tables (Contingency Tables)

Two-way tables show the relationship between two categorical variables.

Two-Way Table: Gender and Preference

A survey asked 200 people about their preferred coffee type.

	Espresso	Latte	Drip	Row Total
Male	35	25	40	100
Female	20	45	35	100
Column Total	55	70	75	200

Reading the table:

35 males prefer espresso
45 females prefer latte
70 people total prefer latte

Joint, Marginal, and Conditional Distributions

Distributions from Two-Way Tables

Using the coffee preference table:

Joint Distribution (proportion in each cell):

	Espresso	Latte	Drip
Male	35/200 = 0.175	0.125	0.200
Female	0.100	0.225	0.175

Marginal Distribution (row or column totals):

Male: 100/200 = 50%
Female: 100/200 = 50%
Espresso: 55/200 = 27.5%

Conditional Distribution (given one variable):

Among males, what % prefer latte? 25/100 = 25%
Among females, what % prefer latte? 45/100 = 45%

Insight: Females are much more likely to prefer lattes (45% vs 25%)!

Common Mistakes to Avoid

1. Overlapping Classes

❌ Wrong: 50-60, 60-70, 70-80 (where does 60 go?)

✅ Correct: 50-59, 60-69, 70-79 OR use “less than” notation like 50 to under 60, 60 to under 70

2. Unequal Class Widths

Unless there’s a specific reason, keep class widths equal for easier interpretation.

3. Too Many or Too Few Classes

Too few: Lose important patterns
Too many: Table becomes as confusing as raw data

4. Missing the “Other” Category

For categorical data, include an “Other” category if responses don’t fit existing categories.

5. Not Including Totals

Always include row and column totals—they’re essential for calculations.

When to Use Each Table Type

Table Type	Use When
Simple frequency	Summarizing one categorical variable
Grouped frequency	Summarizing numerical data with many values
Relative frequency	Comparing groups of different sizes
Cumulative frequency	Finding percentiles, proportion above/below a value
Two-way table	Exploring relationship between two categorical variables

Real-World Application

Hospital Emergency Room Data

An ER wants to understand patient arrival patterns.

Data: 500 patient arrivals over one week

Frequency Table by Time of Day:

Time Period	Patients	Rel. Freq	Cum. Freq
12am-4am	45	9.0%	45
4am-8am	55	11.0%	100
8am-12pm	120	24.0%	220
12pm-4pm	95	19.0%	315
4pm-8pm	110	22.0%	425
8pm-12am	75	15.0%	500

Insights for staffing decisions:

Peak hours: 8am-12pm (24%) and 4pm-8pm (22%)
Lowest: 12am-4am (9%)
By 4pm, 63% of daily patients have arrived

Two-Way Table by Severity:

	Minor	Moderate	Severe	Total
Weekday	180	100	45	325
Weekend	95	55	25	175
Total	275	155	70	500

Insight: Weekends have proportionally similar severity distribution (54.3% minor vs 55.4% weekday minor).

Summary

In this lesson, you learned:

Frequency tables organize data by counting occurrences
Relative frequency expresses counts as proportions (sum to 1.00)
Grouped frequency tables use classes for numerical data
Class width = Range / Number of Classes
Cumulative frequency shows running totals (useful for percentiles)
Two-way tables show relationships between categorical variables
Joint, marginal, and conditional distributions extract different information

Practice Problems

1. Create a grouped frequency table for these ages (in years): 23, 45, 31, 28, 52, 38, 41, 29, 35, 47, 33, 26, 44, 39, 31, 48, 27, 36, 42, 34

Use 5 classes starting at 20.

2. Using your table from Problem 1, add relative frequency and cumulative frequency columns.

3. In a two-way table, 120 out of 200 urban residents and 60 out of 150 rural residents support a policy. Calculate:

Joint distribution of “Urban + Support”
Conditional distribution: % support among urban residents
Marginal distribution: % who support overall

Click to see answers

1. Range = 52 - 23 = 29, Width = 29/5 ≈ 6 (use 6)

Class	Frequency
20-25	1
26-31	6
32-37	4
38-43	4
44-49	4
50-55	1
Total	20

Class	Freq	Rel. Freq	Cum. Freq	Cum. Rel. Freq
20-25	1	0.05	1	0.05
26-31	6	0.30	7	0.35
32-37	4	0.20	11	0.55
38-43	4	0.20	15	0.75
44-49	4	0.20	19	0.95
50-55	1	0.05	20	1.00

Total = 200 + 150 = 350
Total support = 120 + 60 = 180
Joint (Urban + Support) = 120/350 = 34.3%
Conditional (Support | Urban) = 120/200 = 60%
Marginal (Support) = 180/350 = 51.4%

Next Steps

Now that you can organize data in tables:

Measures of Central Tendency - Summarize with numbers
Data Visualization - Turn tables into charts
Percentiles and Quartiles - Use cumulative frequencies

Why Summarize Data in Tables?

Frequency Tables for Categorical Data

Basic Frequency Table

Adding Relative Frequency

Frequency Tables for Numerical Data

Creating Classes

Class Boundaries and Midpoints

Cumulative Frequency Tables

Cumulative Frequency

Cumulative Relative Frequency

Two-Way Tables (Contingency Tables)

Joint, Marginal, and Conditional Distributions

Common Mistakes to Avoid

1. Overlapping Classes

2. Unequal Class Widths

3. Too Many or Too Few Classes

4. Missing the “Other” Category

5. Not Including Totals

When to Use Each Table Type

Real-World Application

Summary

Practice Problems

Next Steps

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