Frequency tables &
Graphical techniques

SOC 221 • Lecture 2

Victoria Sass

Wednesday, June 25, 2025

Frequency tables

What do we do with data?

Apply some STATISTICS to answer interesting questions…

Descriptive statistics

Descriptive statistics
Procedures that help us organize
summarize, and describe the
distribution of a variable.

Distribution
The list of the entire set of observed
values of a variable, indicating all values
and how often those values occur.

• Descriptive statistics allow us to summarize information in a distribution more efficiently
• The type of descriptive statistics we can use to summarize a variable depends on how the variable is measured (i.e. its LEVEL OF MEASUREMENT)

Person #	birth_year	gender	ethnicity	race
1	1992	1	1	5
2	1993	2	0	5
3	1995	2	0	2
4	1980	2	0	3
5	1991	1	1	5
6	1975	4	1	1
7	1960	1	0	6
8	1952	1	0	5
9	2000	3	0	3
10	1990	1	1	2
11	1993	2	1	4
12	1992	3	0	4

Each column of our dataset contains the distribution of a different variable

Frequency tables: overview

Purpose of a frequency table

• Frequency distribution table = descriptive tool to display the frequency of cases with various values/scores on the variable of interest.
• Often present frequencies in multiple ways
  • raw frequencies
  • relative frequencies (percentages)
  • cumulative frequencies
  • cumulative percentages
  • etc.

Frequency table: step 1

Record raw data

Movie ratings (scale from 0 to 10 for 110 viewers)

>   [1]  5  8  3  6  5  8  3  7  9 10  6  8  7  4  6  5
>  [17]  8  7  4  6  5  3  6  1  8  6  8  5  6  6  8  9
>  [33]  6  9  6  5  9  9  6  5  8  5  8  4  8  6  7  4
>  [49]  8  6  8  7  6  8  5  7  6  9  6 10  4  8  6  8
>  [65]  6  5  4  7  8  5  7  4  5  5  9  6  6  7  5  3
>  [81] 10  3  5  7  9 10  6  7  4 10  6  9  8  7  8  5
>  [97]  7  4 10  6  7  4  2  3  8  5  4  7  7  7

Frequency table: step 2

Count the number of cases for each category

0
1	/
2	/
3	///// /
4	///// //// /
5	///// ///// ///// //
6	///// ///// ///// ///// ///
7	///// ///// ///// //
8	///// ///// ///// ////
9	///// ////
10	///// /

Frequency table: step 3

Place in raw frequency table

Table 1: Raw Frequency Distribution of Movie Ratings

Movie Rating (score)	Frequency (f)
0	0
1	1
2	1
3	6
4	11
5	17
6	23
7	17
8	19
9	9
10	6
Total (N)	110

Descriptive title

First column: all possible values (sometimes labeled with a variable symbol, e.g. x )

Second column: number of times each value appears in the data

N = total number of cases in the data

Frequency table: step 4:

Add a percentage (relative frequency) column

Table 1: Raw Frequency Distribution of Movie Ratings

Movie Rating (score)	Frequency (f)	Percent (%)
0	0	0
1	1	0.9
2	1	0.9
3	6	5.4
4	11	10
5	17	15.5
6	23	20.9
7	17	15.5
8	19	17.3
9	9	8.2
10	6	5.4
Total (N)	110	100

\[ Pct_x = \frac{f_x}{N} (100) \]

\[ Pct_0 = \frac{0}{110} (100) \]

\[ Pct_5 = \frac{17}{110} (100) \]

Percentages always add to 100

Frequency table: step 4.1:

Make comparisons across categories

Table 1: Raw Frequency Distribution of Movie Ratings

Movie Rating (score)	Frequency (f)	Percent (%)
0	0	0
1	1	0.9
2	1	0.9
3	6	5.4
4	11	10
5	17	15.5
6	23	20.9
7	17	15.5
8	19	17.3
9	9	8.2
10	6	5.4
Total (N)	110	100

Comparing relative frequency of different values \[ Ratio = \frac{Pct_x}{Pct_y} = \frac{f_x}{f_y} \]

Ratio of 5s to 10s: \[ \frac{15.5}{5.4} = 2.87 \]

Frequency table: step 4.2:

Make comparisons across groups

Table 1: Frequency Distribution of Movie Ratings by Gender

(a) Females

Movie Rating (score)	Frequency (f)	Percent (%)
0	0	0.0
1	0	0.0
2	0	0.0
3	1	2.0
4	3	6.0
5	5	10.0
6	9	18.0
7	10	20.0
8	12	24.0
9	6	12.0
10	4	8.0
Total (N)	50	100

(b) Males

Frequency (f)	Percent (%)
0	0.0
1	1.7
1	1.7
5	8.3
8	13.3
12	20.0
14	23.3
7	11.7
7	11.7
3	5.0
2	3.3
60	100

Gender ratio of 10s: \[ \frac{8.0}{3.3} = 2.42 \]

Frequency table: step 5

Add a cumulative frequency column

Table 1: Raw Frequency Distribution of Movie Ratings

Movie Rating (score)	Frequency (f)	Percent (%)	Cf
0	0	0	0
1	1	0.9	1
2	1	0.9	2
3	6	5.4	8
4	11	10	19
5	17	15.5	36
6	23	20.9	59
7	17	15.5	76
8	19	17.3	95
9	9	8.2	104
10	6	5.4	110
Total (N)	110	100

Cumulative frequency (Cf) =
Number of cases in the given category and lower categories

Cf in highest category = N

Frequency table: step 6

Add a cumulative percentage column

Table 1: Raw Frequency Distribution of Movie Ratings

Movie Rating (score)	Frequency (f)	Percent (%)	Cf	CPcnt
0	0	0	0	0
1	1	0.9	1	0.9
2	1	0.9	2	1.8
3	6	5.4	8	7.2
4	11	10	19	17.2
5	17	15.5	36	32.7
6	23	20.9	59	53.6
7	17	15.5	76	69.1
8	19	17.3	95	86.4
9	9	8.2	104	94.6
10	6	5.4	110	100
Total (N)	110	100

Cumulative
percentage (C%) =
Percentage of cases in the given category and lower categories

\[ C\%_x = \frac{Cf_x}{N} (100) \]

\[ C\%_4 = \frac{19}{110} (100) \]

C% in highest category = 100

Things to watch out for

Reverse ordering of categories

Frequency Distribution of Number of Statistics Classes Completed

Number of courses	Frequency (f)	Percent (%)	Cf	CPcnt
6	2
5	4
4	8
3	24
2	26
1	18
0	3
Total (N)	85

Things to watch out for

Reverse ordering of categories

Frequency Distribution of Number of Statistics Classes Completed

Number of courses	Frequency (f)	Percent (%)	Cf	CPcnt
6	2	2.35	85	100.00
5	4	4.71	83	97.65
4	8	9.41	79	92.94
3	24	28.24	71	83.53
2	26	30.59	47	55.29
1	18	21.18	21	24.71
0	3	3.53	3	3.53
Total (N)	85	100

Things to watch out for

Level of measurement makes some columns inappropriate

Frequency Distribution of College Majors

Major	Frequency (f)	Percent (%)	Cf	CPcnt
Arts	61
Humanities	385
Social Science	425
Business	265
Physical Sciences	194
Other	10
Total (N)	1340

Level of measurement?

Things to watch out for

Level of measurement makes some columns inappropriate

Frequency Distribution of College Majors

Major	Frequency (f)	Percent (%)	Cf	CPcnt
Arts	61	4.55
Humanities	385	28.73
Social Science	425	31.72
Business	265	19.78
Physical Sciences	194	14.48
Other	10	0.75
Total (N)	1340	100

Frequency or % “above” or “below” makes no sense without inherent ordering

Things to watch out for

What seems strange here?

Frequency Distribution of Satisfaction with SOCI 708

Response	Frequency (f)	Percent (%)	Cf	CPcnt
Very Dissatisfied	7	9.33	7	9.33
Somewhat Dissatisfied	22	29.33	29	38.67
Somewhat Satisfied	20	26.67	49	65.33
Very Satisfied	8	10.67	57	76.0
No Answer	18	24	75	100.00
Total (N)	75	100

Things to watch out for

Missing information

Frequency Distribution of Satisfaction with SOCI 708

Response	Frequency (f)	Percent (%)	Cf	CPcnt
Very Dissatisfied	7	9.33	7	9.33
Somewhat Dissatisfied	22	29.33	29	38.67
Somewhat Satisfied	20	26.67	49	65.33
Very Satisfied	8	10.67	57	76.0
No Answer	18	24	75	100.00
Total (N)	75	100

Question: Was a majority at least somewhat dissatisfied with the course (somewhat dissatisfied or worse)?

Things to watch out for

Missing information

Frequency Distribution of Satisfaction with SOCI 708

Response	Frequency (f)	Percent (%)	Cf	CPcnt
Very Dissatisfied	7	12.28	7	12.28
Somewhat Dissatisfied	22	38.60	29	50.88
Somewhat Satisfied	20	35.09	49	85.96
Very Satisfied	8	14.04	57	100
No Answer	18	24	75	100.00
Valid (N)	57	100

Question: Was a majority at least somewhat dissatisfied with the course (somewhat dissatisfied or worse)?

Best practice: Calculate all statistics based on information from those cases with valid (known) information on the variables of interest.

Practice

The following table contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012.

Deaths Worldwide due to Earthquakes

Year	Total Number of Deaths
2000	231
2001	21,357
2002	11,685
2003	33,819
2004	228,802
2005	88,003
2006	6,605
2007	712
2008	88, 011
2009	1,790
2010	320,120
2011	21,953
2012	768
Total (N)	823,856

Answer the following questions:

1. What is the frequency of deaths measured from 2006 through 2009? (inclusive)
   
2. What percentage of deaths occurred after 2009?
   
3. What is the relative frequency of deaths that occurred in 2003 or earlier?
   
4. What is the percentage of deaths that occurred in 2004?
   
5. What kind of data are the numbers of deaths?
   
6. The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers?
   

Break!

Graphical techniques

Graphical Techniques
Descriptive tools to
display the distribution
of variables, or the
association between
variables, using
pictures or figures.

• Huge topic that people spend their lifetimes perfecting

Today’s goals

• Brief intro to basic types
• Basic guidelines for responsible graphing (given great potential to confuse or mislead)

5 rules for responsible graphs

1. Choose the right kind of graph
   • Level of measurement
   • Bar charts, histograms, line charts etc.
2. Clearly label the nature of the information
   • Title and axis labels are standard
3. Focus on your perspective
   • Watch for angles, axis dimensions that skew results
4. Keep it simple
   • Avoid extraneous, distracting information
   • Avoid the use of shapes that do not help convey information
5. Be objective!

Rule 1: Choose the right kind of graph

Nominal & ordinal variables

Bar chart/graph: % or number of cases in each category

• Summarizes the distribution (relative size of the categories)
• Questions:
  • Biggest category?
  • Smallest category?
  • How much bigger is the biggest category from the 2nd biggest category?

Rule 1: Choose the right kind of graph

Nominal & ordinal variables

Bar chart/graph: % or number of cases in each category

Categories do not necessarily need to add up to 100%

• This chart likely contains responses to several different questions/items.
  • Did you drink on Monday?
  • Did you drink on Tuesday?
  • Etc.
• Level of measurement of the variables produced by these items?
  • Nominal
• This type of variable is called a dichotomous (2 values) or dummy (0, 1) variable.

Rule 1: Choose the right kind of graph

Interval variables

Histogram: summarizes the number of cases with each value (or range) on a variable

• Usually used for continuous variables (this is why the bars touch)
• All bars add up to 100% of individuals

Histograms tell us about the SHAPE of a distribution

• Center:
  • Most common value or values
  • Where the cases tend to be clustered/centered

• Spread:
  • Is there a lot of diversity (cases spread out across the values of the variable)…
  • Or are most cases clustered within a few values

• Symmetry vs. Skewness

Rule 1: Choose the right kind of graph

Interval variables

Histogram: summarizes the number of cases with each value (or range) on a variable

• Usually used for continuous variables (this is why the bars touch)
• All bars add up to 100% of individuals

• What can you say about the shape of this distribution?

Rule 1: Choose the right kind of graph

Time-series data

Time plots: Shows us trends and discontinuity (unusual periods)

• Time always on the horizontal (x) axis
• Value of the variable always on the vertical (y) axis.

5 rules for responsible graphs

1. Choose the right kind of graph
   • Level of measurement
   • Bar charts, histograms, line charts etc.
2. Clearly label the nature of the information
   • Title and axis labels are standard
3. Focus on your perspective
   • Watch for angles, axis dimensions that skew results
4. Keep it simple
   • Avoid extraneous, distracting information
   • Avoid the use of shapes that do not help convey information
5. Be objective!

Rule 2: Clearly label the nature of the information

5 rules for responsible graphs

1. Choose the right kind of graph
   • Level of measurement
   • Bar charts, histograms, line charts etc.
2. Clearly label the nature of the information
   • Title and axis labels are standard
3. Focus on your perspective
   • Watch for angles, axis dimensions that skew results
4. Keep it simple
   • Avoid extraneous, distracting information
   • Avoid the use of shapes that do not help convey information
5. Be objective!

Rule 3: Focus on your perspective

Rule 3: Focus on your perspective

5 rules for responsible graphs

1. Choose the right kind of graph
   • Level of measurement
   • Bar charts, histograms, line charts etc.
2. Clearly label the nature of the information
   • Title and axis labels are standard
3. Focus on your perspective
   • Watch for angles, axis dimensions that skew results
4. Keep it simple
   • Avoid extraneous, distracting information
   • Avoid the use of shapes that do not help convey information
5. Be objective!

Rule 4: Keep it simple

5 rules for responsible graphs

1. Choose the right kind of graph
   • Level of measurement
   • Bar charts, histograms, line charts etc.
2. Clearly label the nature of the information
   • Title and axis labels are standard
3. Focus on your perspective~~
   • Watch for angles, axis dimensions that skew results
4. Keep it simple
   • Avoid extraneous, distracting information
   • Avoid the use of shapes that do not help convey information
5. Be objective!

Rule 5: Be objective

Example of axis truncation

What does this chart imply about divorce rates?

Note the difference in the X and Y axis.

Rule 5: Be objective

No consistent y-axis

Graphic Presented by Congressman Jason Chaffetz (R-UT) During Meeting About Planned Parenthood Funding. September 29, 2015

Rule 5: Be objective

Purposefully misleading

Practice

Seeing is believing 👀

1. Find a data visualization on the internet that you find to be misleading or confusing. Send the link to our class slack channel #badviz.

If you need help sourcing bad data visualizations, try the subreddit r/dataisugly

2. Find a data visualization on the internet that you think conveying information efficiently and follows all five (5) responsible graph rules. Send the link to our class slack channel #goodviz.

If you need help sourcing good data visualizations, try the subreddit r/dataisbeautiful

Note: If someone has already posted the visualization you wanted to share, just pick another one!

Fin!