Frequency Tables and Dot Plots - Sixth Grade Mathematics
Organizing raw observational data into structured frequency tables and visual dot plots reveals patterns, clusters, peaks, and spread in numerical distributions.
Raw Data to Frequency Tables
When data is collected, it initially appears as an unstructured list of numbers called raw data. A frequency table organizes this data by listing each distinct value alongside tally marks and its frequency (the number of times that value occurs).
Raw Data (Test Scores out of 10 for 20 students):
8, 9, 7, 10, 8, 6, 7, 8, 9, 8, 10, 5, 7, 8, 9, 6, 8, 7, 9, 8
Constructing the Frequency Table:
+-------+--------------+-----------+
| Score | Tally | Frequency |
+-------+--------------+-----------+
| 5 | | | 1 |
| 6 | || | 2 |
| 7 | |||| | 4 |
| 8 | |||| | | 6 |
| 9 | |||| | 4 |
| 10 | || | 2 |
+-------+--------------+-----------+
| Total | | 20 |
+-------+--------------+-----------+
Verifying that the sum of the frequency column equals the total sample size (N = 20) ensures that no data points were omitted or double-counted.
Dot Plots (Line Plots)
A dot plot (also historically called a line plot) is a visual display where each data value is plotted as a single dot (or an X) placed vertically above its position on a continuous horizontal number line.
Dot Plot of the 20 Test Scores:
*
*
* * *
* * *
* * * * *
* . * . * . * . *
+---+----+---+----+---+----+---+----+---+----+
4 5 6 7 8 9 10 11 12 13 14
Test Scores
Every individual dot represents exactly one observation. In the dot plot above, there is 1 dot at 5, 2 dots at 6, 4 dots at 7, 6 dots at 8, 4 dots at 9, and 2 dots at 10.
Analyzing Distribution Features
Visualizing data with a dot plot highlights key structural features:
1. Peak (Mode): The value with the tallest stack of dots, representing the most frequent observation. In our plot, the peak is at 8.
2. Clusters: Groups where data points gather closely together (here, between 7 and 9).
3. Gaps: Empty spaces on the number line where no data values occur.
4. Outliers: Individual values that sit noticeably isolated from the main body of data.
5. Spread (Range): The overall span from the minimum value to the maximum value (10 - 5 = 5).
Distribution Shapes:
- Symmetrical: The left half and right half mirror each other around a central peak (like the bell-shaped plot above).
- Skewed Right: The tail of low-frequency dots stretches out toward the higher values on the right.
- Skewed Left: The tail stretches out toward the lower values on the left.
Relative Frequency and Percentages
Relative frequency expresses the proportion of the total data set represented by each value: Relative Frequency = Frequency / Total Count (N). For a score of 8: Relative Frequency = 6 / 20 = 0.30 = 30%. Thirty percent of the students scored an 8.
Exercises
Exercise 1. The following list represents the number of books read by 15 students over the summer: 3, 1, 4, 2, 3, 5, 2, 3, 0, 4, 3, 2, 6, 1, 3. (a) Create a frequency table for this data; (b) What is the total count of students; (c) What is the peak (most frequent value); (d) What fraction and percentage of students read exactly 3 books?
Exercise 2. A dot plot displays the number of siblings reported by students: 0 siblings has 4 dots, 1 sibling has 7 dots, 2 siblings has 5 dots, 3 siblings has 3 dots, and 6 siblings has 1 dot. (a) How many total students were surveyed? (b) Identify any gaps in the data; (c) Identify any potential outliers; (d) How many students have 2 or more siblings?
Exercise 3. Describe the difference between a symmetrical dot plot and a skewed dot plot.
Solutions
Solution to Exercise 1. For (a), the frequency table: 0 books: 1; 1 book: 2; 2 books: 3; 3 books: 5; 4 books: 2; 5 books: 1; 6 books: 1. For (b), the total count is 1 + 2 + 3 + 5 + 2 + 1 + 1 = 15 students. For (c), the peak is 3 books with a frequency of 5. For (d), 5 out of 15 students read 3 books: fraction = 5/15 = 1/3; percentage = 33 1/3% (or approx. 33.3%).
Solution to Exercise 2. For (a), total students = 4 + 7 + 5 + 3 + 1 = 20 students. For (b), there is a gap at 4 and 5 siblings where 0 dots appear. For (c), the value of 6 siblings is an outlier because it is separated from the rest of the cluster (0 to 3) by a gap of two empty values. For (d), count students with 2, 3, or 6 siblings: 5 + 3 + 1 = 9 students.
Solution to Exercise 3. A symmetrical dot plot is balanced, where values on either side of the central peak decrease at similar rates resembling a bell shape. A skewed dot plot is asymmetrical, having a peak shifted toward one end and a long, thin tail of sparse data stretching out toward the opposite end (skewed right if the tail is to the right; skewed left if the tail is to the left).