Notes on Frequency Tables and Distributions

Frequency Table Basics

A frequency table displays the number of occurrences of each score (the frequency) in a dataset. It shows how many times each value appears (e.g., 1, 2, 3, 4, 5) for the variable.
In a frequency table, you typically include:
- The raw scores (the actual values)
- The frequencies (counts of each score)
- The percentages (proportions converted to percent) for easy interpretation.
In psychology, frequency tables are valuable because they:
- Summarize data concisely
- Make interpretation easy
- Help identify patterns, trends, or changes (e.g., symptoms before/after an intervention, changes over time)
Frequency tables are a foundation for understanding distributions and for planning subsequent visualizations or analyses.

Proportions and Percentages

Proportions and percentages serve the same purpose: they describe the relative size of each category with respect to the total.
Proportion (fraction of the total):
- $\text{Proportion} = \frac{f}{N}$
- where $f$ is the frequency for a given score and $N$ is the total number of observations.
Percentage (proportion expressed as a percent):
- $\text{Percentage} = \text{Proportion} \times 100 = \frac{f}{N} \times 100$
Example:
- If 30 out of 40 responses have a score of 1, then
- $\text{Proportion} = \frac{30}{40} = 0.75$
- $\text{Percentage} = 0.75 \times 100 = 75\%$
Proportions and percentages convey the same information; percentages are typically easier to interpret at a glance.
Note: If you state that a certain percentage scored a value (e.g., 38%), you are describing the distribution of the entire sample.

How to Make a Frequency Table

Dataset example (variable with scores 1–5): values might be something like a stress level measure.
Steps to create a frequency table:
- Step 1: List the possible scores (e.g., 1, 2, 3, 4, 5).
- Step 2: Count how many times each score occurs in the dataset.
- Step 3: Compute the proportion for each score:
- $\text{Proportion}k = \frac{fk}{N}$ where $f_k$ is the frequency of score $k$ and $N$ is the total number of observations.
- Step 4: Compute the percentage for each score:
- $\text{Percentage}k = \text{Proportion}k \times 100 = \frac{f_k}{N} \times 100$
Manual counting vs. Excel:
- Manual counting: count occurrences of each score (e.g., how many 1s, how many 2s, etc.).
- Excel COUNTIF method:
- For a given range containing the dataset, use
  - $f_k = \text{COUNTIF}(\text{range}, k)$
  - Example: $f_1 = \text{COUNTIF}(\text{range}, 1)$
- Repeat for each score (1, 2, 3, 4, 5).
After counting, you can compute the proportions and percentages as above. When copying formulas in Excel, paste values if you want to fix the numbers rather than keep links to the original data.
The total of the frequencies is $N$ , and the sum of all percentages should be $100\%$ (subject to rounding).

Interpreting the Data from a Frequency Table

Example interpretation (stress levels):
- Suppose the distribution shows that 38% scored 2 (low stress).
- Given the scale mapping: 1 = very low, 2 = low, 3 = neutral, 4 = high, 5 = very high,
- Most people scored low on stress (score 2).
- This supports conclusions such as: a majority of respondents report relatively low stress levels.
When interpreting, relate categories to the construct and consider the implications for therapy, interventions, or further study.

Graphs for Frequency Distributions

Types of graphs and when to use them:
- Histogram: best for interval/ratio data (continuous, numeric scores). It shows frequency along the vertical axis and score ranges on the horizontal axis. Bars touch (continuous distribution).
- Bar graph: best for nominal or ordinal data (categories with no natural numeric order or with categories that are ordered). Bars are separated (discrete categories).
- Modified histogram: a histogram where each box represents a frequency (one box = one occurrence). It can make frequency counts easier to read.
- Frequency polygon: an alternative to histogram for interval/ratio data; connect the midpoints of the histogram bars to form a polygon.
- Smooth curve: used with interval/ratio data to show a clean, continuous distribution; helpful to visualize a bell-shaped (normal) distribution.
Guidelines:
- Use histogram for interval/ratio data (e.g., test scores, age in bins).
- Use bar graph for nominal/ordinal data (e.g., country choices, Likert-scale categories).
- If you have grouped data (bins) or multiple categories, a histogram (or modified histogram) can be appropriate.
Excel-specific notes:
- When you insert a histogram in Excel, you might get a bar chart by default; you can format the series and set gap width to 0 to resemble a histogram, and adjust category labels to reflect the actual category names (e.g., Levels of stress) instead of raw numbers.
- For nominal data (categories like Japan, Korea, Philippines, China), a bar graph is typically used, with each bar representing a category and height representing frequency or percentage.
Example in practice:
- Nominal data: country preference (Japan, Korea, Philippines, China) -> use a bar graph with frequencies or percentages per country.
- Ordinal data: level of stress (1–5) -> can be treated as interval for some purposes; histogram can be used if you collapse into bins, or a bar graph if you treat each level as a category.
Important caveat: For nominal data, the x-axis labels are categories (not numeric values); for interval/ratio data, the x-axis represents numeric scores or bin ranges.

Handling Different Types of Data and Multiple Responses

Data types by level of measurement:
- Nominal: categories with no intrinsic order (e.g., country, type of phobia).
- Ordinal: categories with order but not equal intervals (e.g., Likert scales).
- Interval/Ratio: numeric scales with meaningful intervals; ratio has a true zero (e.g., weight, IQ scores).
For interval/ratio data, you can create histograms or frequency polygons to visualize the distribution.
For nominal/ordinal data, bar graphs (or sometimes paragraph-like displays) are more appropriate.
Multiple responses per participant: when a respondent can endorse more than one cause (e.g., different anxiety factors), you cannot simply sum frequencies across causes because causes may be non-exclusive and not mutually exclusive measures of the same construct.
- You must consider the total number of respondents (N) when calculating proportions for each cause.
- For each cause, compute the proportion relative to N: $\text{Proportion}{\text{cause}} = \frac{f{\text{cause}}}{N}$ and the corresponding percentage: $\text{Percentage}{\text{cause}} = \text{Proportion}{\text{cause}} \times 100$
Grouping nominal data: you can group related categories if it makes theoretical or practical sense (e.g., grouping various phobias into a broader “specific phobia” category or grouping academic factors like academic performance and ineffective study habits under a scholastic/school-related construct). When grouping, you sum frequencies for the aggregated category and compute the corresponding proportion/percentage.
Age or score intervals: interval data can be shown in grouped form (e.g., age bins like 13–17, 18–22, 23–24). The frequency table can then reflect these intervals rather than individual values.
Caution about interpretation:
- Data limitations can affect interpretation (sample size, rounding, non-response).
- In psychology, it’s common to interpret distributions in the context of theoretical constructs (e.g., stress levels) and to connect findings to prior lectures on central tendency and variability.

Additional Concepts and Examples Mentioned

Relationship to central tendency (referenced as a later topic): frequency distribution informs calculations of the mean, median, and mode, and supports understanding of where most observations lie.
Terminology clarifications:
- Frequency: the count of occurrences for a score.
- Proportion: the fraction of the total corresponding to a given score.
- Percent: proportion expressed as a percent.
Example of a normal/bell-curve intuition: many values cluster around the mean (e.g., IQ scores with a mean of 100), with fewer observations at the extremes (e.g., 60 or 140).
Activity for practice (assignment): describe why psychologists create graphs and produce graphs for provided data on two samples (A and B) on a half-page crosswise sheet.

Quick Reference: Key Formulas

Proportion for score k: $\text{Proportion}k = \frac{fk}{N}$
Percentage for score k: $\text{Percentage}k = \text{Proportion}k \times 100 = \frac{f_k}{N} \times 100$
Sum checks: the sum of all frequencies equals the total $N$ , and the sum of all percentages equals 100\% (subject to rounding).

Practical Takeaways

Use frequency tables to summarize data concisely and identify patterns quickly.
Choose the right graph type based on data level: histogram/modified histogram/polygon/smooth curve for interval/ratio data; bar graph for nominal/ordinal data.
When data include multiple responses per participant, avoid simply summing across categories; base proportions on the total number of respondents and describe each category accordingly.
Always connect the presentation to the construct under study (e.g., level of stress) and provide a meaningful interpretation grounded in the context of psychology.

Questions to Consider (from the activity)

Why do psychologists create graphs?
How would you graph data for the given A and B datasets? (Think about the data type and the appropriate graph type, and be prepared to justify your choice.)