Mathematics • Year 10 • Unit 4 • Lesson 3
Frequency Tables and Dot Plots, Skill Drill
Build fluency with the two simple displays from Lesson 3: frequency tables (how often each value occurs) and dot plots (each value shown as a dot above a number line). Practise reading off frequency, identifying mode, clusters, gaps and outliers, and checking that frequencies sum to the total.
1. I do, fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. A teacher records the number of children in each of 20 students' households:
2, 3, 2, 1, 4, 2, 3, 1, 2, 5, 3, 2, 4, 2, 3, 1, 2, 3, 2, 4.
Build a frequency table and identify the mode.
Step 1, List the distinct values from smallest to largest.
Distinct values: 1, 2, 3, 4, 5.
Reason: a frequency table needs one row per distinct value.
Step 2, Tally and count each value.
1 → ||| (3) | 2 → |||| |||| (8) | 3 → |||| (5) | 4 → ||| (3) | 5 → | (1)
Reason: Lesson 3 Key Term, "frequency = the number of times a particular value occurs in a data set".
Step 3, Sum-check.
3 + 8 + 5 + 3 + 1 = 20 ✓ (matches the 20 students).
Reason: frequencies must always sum to the total number of data values.
Step 4, Identify the mode.
Highest frequency is 8 (for value 2). Mode = 2.
Reason: the mode is the most-frequent value, the tallest bar/stack.
Answer: Frequencies {1:3, 2:8, 3:5, 4:3, 5:1}. Mode = 2 children.
2. We do, fill in the missing dot plot
Same idea as Section 1, but you build the display. Fill in each blank line. 4 marks
Problem. A die is rolled 15 times. The results are:
4, 6, 3, 4, 1, 5, 4, 2, 4, 6, 3, 4, 5, 2, 4.
Build a frequency table and describe the dot plot's shape using the words cluster, gap and outlier.
Step 1, Distinct values (smallest to largest):
______, ______, ______, ______, ______, ______
Step 2, Frequencies (tally each value):
1 → ____ | 2 → ____ | 3 → ____ | 4 → ____ | 5 → ____ | 6 → ____
Step 3, Sum-check (must equal 15):
Sum = ____ → matches 15? ____ (Y / N)
Step 4, Mode. The tallest stack on the dot plot is at value ______, so mode = ______.
Step 5, Describe the shape. The data shows a cluster around _________ and a small dip at value(s) _________. There are/are not (circle one) outliers, because ______________________________.
3. You do, independent practice
Show your working in the space under each problem. The first four are foundation (read or build a small table). The middle two are standard (read a dot plot). The last two are extension (use the lesson's misconceptions).
Foundation, build and read frequency tables
3.1 A frequency table for the data 3, 3, 4, 4, 4, 5, 5 shows frequency of 4 = ? 1 mark
3.2 Build a frequency table for the test marks (out of 10):
7, 8, 7, 9, 10, 7, 8, 9, 7, 8. Then state the mode. 1 mark
3.3 A frequency table shows: value 5 → 3, value 6 → 6, value 7 → 8, value 8 → 3. How many data values are there in total? 1 mark
3.4 A dot plot has 4 dots above 12, 7 dots above 13, 5 dots above 14, and 1 dot above 20. State (a) the total number of data values, and (b) the mode. 1 mark
Standard, describe shape from a dot plot
3.5 A dot plot of "siblings per student" for 25 students looks like this (number of dots above each value):
0 → 3, 1 → 9, 2 → 7, 3 → 4, 4 → 1, 5 → 0, 6 → 1.
(a) Identify the mode.
(b) Name and locate any cluster, gap and outlier. 2 marks
3.6 Build a frequency table from this dot plot:
2 → 1, 3 → 2, 4 → 5, 5 → 8, 6 → 5, 7 → 2, 8 → 1.
(a) Total data values = ?
(b) Mode = ?
(c) Describe the overall shape in one sentence (use the word symmetric if appropriate). 2 marks
Extension, use the lesson's misconceptions
3.7 A student says: "On a dot plot, the tallest stack is the mean." Using the Lesson 3 misconceptions card, explain why this is wrong and what the tallest stack actually shows. 2 marks
3.8 A student tries to build a "frequency table" for the values 1, 1, 2, 3, 4, 4, 5 by writing classes 0-2, 2-4, 4-6. (a) State two things that are wrong with these classes. (b) Suggest a correct simple frequency table (not grouped) for this data. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (15 die rolls)
Step 1: 1, 2, 3, 4, 5, 6.
Step 2: 1 → 1, 2 → 2, 3 → 2, 4 → 6, 5 → 2, 6 → 2.
Step 3: 1 + 2 + 2 + 6 + 2 + 2 = 15 ✓.
Step 4: tallest stack at value 4 → mode = 4.
Step 5: Cluster around 4 (and broadly 3-5). No major gap; the smallest value (1) appears once but is not far enough from the cluster to be an outlier. There are no outliers.
3.1, Frequency of 4
4 appears three times: frequency = 3.
3.2, Test marks frequency table
{7: 4, 8: 3, 9: 2, 10: 1}. Sum-check: 4 + 3 + 2 + 1 = 10 ✓. Mode = 7.
3.3, Total values
3 + 6 + 8 + 3 = 20 data values.
3.4, Dot plot total and mode
(a) Total = 4 + 7 + 5 + 1 = 17.
(b) Mode = 13 (tallest stack).
3.5, Siblings dot plot
(a) Mode = 1 sibling (tallest stack of 9 dots).
(b) Cluster around 0-3 (where the bulk of the dots sit), gap at value 5 (no dots), outlier at value 6 (one dot far from the main cluster).
3.6, Build frequency table
{2: 1, 3: 2, 4: 5, 5: 8, 6: 5, 7: 2, 8: 1}.
(a) Total = 1 + 2 + 5 + 8 + 5 + 2 + 1 = 24.
(b) Mode = 5.
(c) The distribution is symmetric (mirror-image around the mode of 5).
3.7, "Tallest stack = mean" misconception
The tallest stack is the mode (the most-frequent value), not the mean. The mean is the average of all values and must be calculated: (sum of all values) ÷ (number of values). The mean might fall between any two stacks and might not correspond to any actual data value at all. Lesson 3 calls this out as a direct misconception.
3.8, Wrong "frequency table"
(a) Two problems: (i) the classes overlap at 2 and 4 (where does a value of 2 go, 0-2 or 2-4?); (ii) these are grouped intervals for a tiny ungrouped data set, which is unnecessary. Lesson 3 misconceptions card: "Frequency tables for individual (ungrouped) values do not use class widths at all."
(b) Correct simple frequency table: {1: 2, 2: 1, 3: 1, 4: 2, 5: 1}. Total: 2 + 1 + 1 + 2 + 1 = 7 ✓.