الأسبوع 14 · الوسط الحسابي والوسيط والمنوال لجدول تكراري ذي مجموعات · 3 دروس
3
Mode of a grouped table from the histogram, and comparing the three measures
Mathematics Prep 2 Term 1, Unit 4, Lesson 2
GoalFind the modal class and the mode from the histogram with the crossing lines, and compare the mean, the median and the mode.
New words
histogram · bars with no gaps; the height of each bar is the frequency of its classmodal class · the class with the greatest frequency, the tallest bar when classes have equal lengthmode · the most common value, found from the crossing linescrossing lines · two lines drawn between corners of the modal bar and its neighbours; they cross above the mode
Explanation
A histogram draws each class as a bar with no gaps. The tallest bar is the modal class. In the histogram of the ages (frequencies 5, 9, 12, 10, 4) the modal class is 30–. To find the mode, draw two lines in the modal bar: one from its top left corner to the top left corner of the bar on its right, and one from its top right corner to the top right corner of the bar on its left. They cross at a point; the mode is the value under the crossing. Here the lines cross at 36, so the mode is about 36 years. By proportion: the differences from the neighbours are d₁ = 12 − 9 = 3 and d₂ = 12 − 10 = 2, so the mode is 30 + 10 × 3 ÷ (3 + 2) = 36. The three measures answer different questions. The mean, 34.75, uses every value and is pulled by very large or very small values. The median, 35, is the middle value. The mode, 36, is the most common value. Here they are close, which suggests the ages are spread fairly evenly around the middle. The mode from the picture is approximate, and the modal class is the tallest bar only if all the classes have the same length.
Worked examples
Ages in the classes 10–, 20–, 30–, 40–, 50–60 with frequencies 5, 9, 12, 10, 4. The tallest bar is 30–. d₁ = 12 − 9 = 3 and d₂ = 12 − 10 = 2, so the mode is 30 + 10 × 3 ÷ 5 = 36 years. The mean is 34.75 and the median 35, so all three are close.
Waiting times with frequencies 3, 5, 20, 8, 4 in the classes 0–, 10–, 20–, 30–, 40–50. The modal class is 20–. d₁ = 20 − 5 = 15 and d₂ = 20 − 8 = 12, so the mode is 20 + 10 × 15 ÷ 27 ≈ 25.6 minutes. The mean is (15 + 75 + 500 + 280 + 180) ÷ 40 = 1050 ÷ 40 = 26.25 and the median is 26. The three are again close.
Which measure? A shop owner who wants to stock the most popular shoe size needs the mode. A teacher who wants the typical mark halfway through the class uses the median. If one very rich person joins a group of 10 people, the mean pay rises a lot, but the median hardly changes, so the median describes a typical person better.
Class activity
Build a histogram. Using your class-survey table, draw a histogram on squared paper with the bars touching. Find the modal class and use the crossing lines to find the mode. Compare it with the mean and the median you found earlier.
Practice
1
Classes 0–, 10–, 20–, 30–40 have frequencies 3, 4, 10, 8. Find the modal class and the mode.
