DAT Quantitative Reasoning: Reading Frequency Tables and Cumulative Counts

DAT High-Yield Concepts

Frequency Table Structure: Pairs a data value or category ( $x_i$ ) with its frequency ( $f_i$ ), which represents how many times that exact value occurs.
Total Sample Size ( $n$ ): Found by summing all counts in the frequency column: $n = \sum_{i=1}^{k} f_i$
Locating the Median: Expand the table into an ordered list or use rank formulas: position $\frac{n+1}{2}$ for odd $n$ , or average of positions $\frac{n}{2}$ and $\frac{n}{2} + 1$ for even $n$ .
Cumulative Frequency Tracking: Running totals down the frequency column pinpoint the exact value corresponding to any given rank without rewriting raw data.

Understanding Frequency Tables and Sample Size

A frequency table organizes raw quantitative data into a condensed format. Each entry pairs a discrete value or class interval $x_i$ with its frequency $f_i$ . The total sample size $n$ is determined strictly by summing the frequency column, not by counting the number of rows or summing the data values.

Frequency table of pulse rates; based on the dataset, survey , available in R's MASS package. (Source: ASnieckus · CC BY-SA 3.0 · Wikimedia Commons)

Sample Size Rule: Total sample size

n = \sum f_i

. The sum of data values

\sum x_i

is distinct from the sample size

n

Worked Example 1: Finding Total Sample Size

Problem: A study records the number of post-operative checkups required by dental patients:

Checkups ( $x$ ): 1, 2, 3, 4, 5
Frequency ( $f$ ): 3, 5, 8, 4, 2

Determine the total number of patients evaluated in this study.

Solution: Sum the values in the frequency column: $n = 3 + 5 + 8 + 4 + 2 = 22$ There are 22 patients in total.

Common Exam Trap: Do not sum the data value column ( $1 + 2 + 3 + 4 + 5 = 15$ ). The data values represent response options, whereas frequencies represent count totals.

Expanding Frequency Tables to Locate the Median

The median is the middle observation when data are arranged in ascending order. When working with small frequency tables, expanding the counts back into an ordered list helps identify central tendency clearly.

Worked Example 2: Finding the Median via List Expansion

Problem: A clinic tracks patient satisfaction scores on a scale from 7 to 10:

Score ( $x$ ): 7, 8, 9, 10
Frequency ( $f$ ): 2, 4, 3, 1

Find the median satisfaction score.

Solution:
1. Calculate sample size: $n = 2 + 4 + 3 + 1 = 10$ .
2. Expand into an ordered list: 7, 7, 8, 8, 8, 8, 9, 9, 9, 10.
3. Since $n = 10$ (even), locate the $\frac{10}{2} = 5$ -th and $\frac{10}{2} + 1 = 6$ -th values.
4. Both the 5th and 6th terms equal 8. Average them: $\frac{8 + 8}{2} = 8$ .
The median score is 8.

Common Exam Trap: Avoid finding the median of the unique score values alone ( $\frac{8 + 9}{2} = 8.5$ ). You must account for frequencies when finding position.

Cumulative Frequency and Position Tracking

For larger data sets, expanding full lists is inefficient. Adding a cumulative frequency column ( $cf$ ) tracks running counts and identifies which row contains a target data position.

cumulative frequency absolute relative (Source: SvaRoM · CC BY-SA 3.0 · Wikimedia Commons)

Note: Calculating summary averages across frequency distributions requires the weighted mean $\bar{x} = \frac{\sum x_i f_i}{\sum f_i}$ , which is covered in depth in its dedicated weighted statistics lesson.

Worked Example 3: Running Cumulative Frequencies for Rank Location

Problem: A test records student quiz scores:

Score ( $x$ ): 5, 10, 15, 20
Frequency ( $f$ ): 4, 6, 7, 3

Determine the value of the 12th data point in the ordered dataset.

Solution:
1. Compute cumulative frequencies:
• $x = 5$ : $cf = 4$ (positions 1–4)
• $x = 10$ : $cf = 4 + 6 = 10$ (positions 5–10)
• $x = 15$ : $cf = 10 + 7 = 17$ (positions 11–17)
• $x = 20$ : $cf = 17 + 3 = 20$ (positions 18–20)
2. The 12th position lies in the range 11–17, which corresponds to $x = 15$ .
The value of the 12th data point is 15.

Common Exam Trap: Do not choose score 10 simply because 12 is closer to 10 than 17. The cumulative frequency indicates the upper boundary position for that score category.

Comparison of Representation Methods

Data Representation	Primary Advantage	Method to Find Median
Raw Data List	Shows individual data points directly	Sort in order and pick middle position
Frequency Table	Summarizes large datasets efficiently	Expand list or use total sample size rank
Cumulative Frequency Table	Allows fast direct lookup of rankings/percentiles	Locate row where $cf$ first exceeds median rank

Mnemonic for Frequency Table Analysis — S-C-O-R-E:

Sum frequencies to find total sample size $n$ .
Calculate cumulative frequencies down the table.
Order rows by score/value ascending.
Rank median or target position using $n$ .
Extract value corresponding to target cumulative range.

Critical Thinking Questions

If a frequency table has an odd total sample size $n = 45$ , which exact rank position represents the median, and how is it identified using cumulative frequencies?
A frequency table lists values 2, 4, 6, and 8 with frequencies 3, $k$ , 5, and 2. If the median lies strictly at value 4, what range of positive integer values can $k$ take?
How does adding a single high-value outlier affect the cumulative frequency column compared to its impact on the mean of a dataset?

References

Sources returned by Google Search grounding during lesson verification.

Median of a frequency distribution - Khan Academy — youtube.com
Cumulative Frequency - Statistics Canada — www150.statcan.gc.ca

Reading Frequency Tables and Cumulative Counts

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