Connecting the Dots... · Lesson 3 of 13
Know Your Onions!
“Use onion prices to compare means, ranges, and the information revealed by dot plots.”
• Read and compare a monthly price table. • Calculate minimum, maximum, and range. • Construct and interpret dot plots with repeated values. • Explain what a dot plot preserves and what it loses. • Make qualified comparisons using more than one measure.
Know Your Onions!
Imagine buying onions in two towns throughout a year. One town reaches the highest price, but the other might have a slightly higher typical price. Prices can also be steadier in one town than the other. A useful comparison therefore starts with the complete dataset and considers more than one feature.
| Month | Yahapur (₹/kg) | Wahapur (₹/kg) |
|---|---|---|
| Jan | 25 | 19 |
| Feb | 24 | 17 |
| Mar | 26 | 23 |
| Apr | 28 | 30 |
| May | 30 | 38 |
| Jun | 35 | 35 |
| Jul | 39 | 42 |
| Aug | 43 | 39 |
| Sep | 49 | 53 |
| Oct | 56 | 60 |
| Nov | 59 | 52 |
| Dec | 44 | 42 |
In October, Wahapur is costlier: ₹60 compared with ₹56. In November, Yahapur is costlier: ₹59 compared with ₹52. Counting month-by-month comparisons gives six months when Yahapur is costlier, five when Wahapur is costlier, and June when they are equal. This is useful information, but the count ignores how large each difference is.
Minimum, maximum, and range
The minimum and maximum are the two ends of a numerical dataset. Their difference gives a quick description of its spread. Two places can have similar average prices while experiencing quite different extremes.
The minimum is the smallest recorded value; the maximum is the largest.
The difference between the maximum and minimum values in a dataset.
Problem
Which town’s prices have the greater range?
- 1.Yahapur: minimum 24 and maximum 59, so range = 59 − 24 = ₹35/kg.
- 2.Wahapur: minimum 17 and maximum 60, so range = 60 − 17 = ₹43/kg.
- 3.Wahapur has the larger range. This describes its extremes, not every month-to-month change.
Seeing data with a dot plot
A table keeps the link between a month and its price. A dot plot instead places each observation at its numerical position. If two observations have the same value, place one dot above the other. Stacks preserve the number of occurrences.
A display with one dot for each observation on a numerical axis; repeated values are stacked at the same position.
The axis begins at 10 because all recorded prices lie above 10. On a dot plot we read position, rather than the length of a bar from a baseline. Equal horizontal distances must still represent equal price differences. Each town has twelve dots, including repetitions. Wahapur has two observations at 42; the two towns each have an observation at 39.
Look for where values gather and where they spread apart. A cluster is a group of nearby values. A gap contains no observations. The dot plot reveals that Wahapur has values in the interval 11–20, while Yahapur has none. It also makes minimum and maximum easy to locate.
Problem
Can the onion dot plot tell us Yahapur’s January price?
- 1.It shows the collection of twelve prices and how often each price occurs.
- 2.The individual dots are not labelled with months, so their original month order is lost.
- 3.We must return to the table to identify January’s ₹25/kg. The representation is useful for spread but not for a time-by-time story.
Comparing representative prices
Because both towns have twelve monthly observations, their totals and their arithmetic means rank them in the same order. Calculate the means, but interpret the difference alongside the ranges and the month-by-month comparison.
Problem
Find the arithmetic mean price for each town.
- 1.Yahapur’s prices sum to 458. Mean = 458 ÷ 12 = 38.166… ≈ ₹38.17/kg.
- 2.Wahapur’s prices sum to 450. Mean = 450 ÷ 12 = ₹37.50/kg.
- 3.The equally weighted monthly mean is about ₹0.67/kg higher in Yahapur. This does not make Yahapur costlier in every month.
The conclusion must say which measure it uses. “Yahapur has the higher mean monthly price” and “Wahapur has the larger range” can both be true. Later, the medians will provide another comparison. The monthly mean used here gives every month equal weight; it is not a calculation weighted by quantities of onions purchased.
Do seasons affect prices? How do prices vary across shops within a town? What other products show similar patterns? The table can suggest these questions, but does not contain enough information to establish their causes.
The highest single price does not identify the higher mean. A larger range does not necessarily mean a larger mean. Choose a description that matches the question.
Quiz
Wahapur’s minimum is 17 and maximum is 60. Its range is:
What do two stacked dots at 42 mean?
Which information is lost in an unlabelled dot plot of monthly prices?
Which statement is supported by the mean calculations?
Which town has the larger range?
Practice Problems
- Identify the months when Wahapur is costlier.
- How much more expensive is Yahapur in November?
- Find the mean and range of prices 20, 20, 25, 30, 45.
- Explain why two towns can have similar means but different ranges.
- Draw a dot plot for Yahapur and state what you can and cannot read from it.
- Does the onion table prove that seasons caused the price changes?
April, May, July, September, and October. June is equal; Yahapur is costlier in the other six months.
Key Takeaways
• Keep units attached to price observations. • Minimum and maximum identify the extremes. • Range describes the distance between the extremes. • Each dot represents one observation, including repeats. • A dot plot shows distribution but may lose chronological order. • Compare centre, spread, and context before making a conclusion.