Chapter 13 ยท Introduction

Chapter 13 โ€” Statistics

Results of a board exam, rainfall over a season, scores in a tournament: real data come as long lists, and long lists are summarised in tables of classes. This chapter shows how to find the three "averages" (mean, mode and median) when all you have is a grouped table.

A school publishes the marks of its $400$ Class 10 students in a table: $87$ students scored between $0$ and $20$, $142$ between $20$ and $40$, and so on. A parent asks three natural questions. What is the average mark? Which score range is the most common? What mark separates the top half of the class from the bottom half? The table has thrown away the individual marks, yet each question can still be answered, to a very good approximation.

You already know the three averages for a short list from earlier classes: the mean (add and divide), the mode (the most frequent value) and the median (the middle value). In this chapter you extend each of them to grouped data. The key trick is to replace every class by its midpoint, the class mark, and to count frequencies as weights. Section 13.1 finds the mean, Section 13.2 the mode, and Section 13.3 the median, and shows a short relation that connects all three.

Raw data in a long listโ†’Grouped table: classes, class marks, frequenciesโ†’Mean $\bar x=\dfrac{\sum f_ix_i}{\sum f_i}$โ†’Modal class and modeโ†’Cumulative frequency, median class and median
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • find the class mark and class size of a class, and make classes continuous when they are written with gaps;
  • find the mean of grouped data by the direct method, the assumed-mean method and the step-deviation method, and explain why they agree;
  • find the modal class and the mode of grouped data with the formula $l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}h$;
  • build a cumulative frequency column, locate the median class and find the median with $l+\dfrac{\frac n2-cf}{f}h$;
  • use $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$ as a check, and choose the average that best describes a situation.
Teaser โ€” One Big Score Changes the Mean, Not the Median

A batter's scores in nine innings are shown as dots. Slide the tenth score from $0$ to $200$ (or press Play). The mean chases the new score, while the median hardly moves and the mode stays at the repeated score. Each average answers a different question, and choosing the right one is part of the skill of statistics.

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meanโ€”
medianโ€”
modeโ€”
sorted scoresโ€”

Sections in this chapter

13.1

Mean of Grouped Data

Use class marks and frequencies to find the mean by the direct method, the assumed-mean method and the step-deviation shortcut.

$\bar x=\dfrac{\sum f_ix_i}{\sum f_i}$ ยท $\bar x=a+\dfrac{\sum f_id_i}{\sum f_i}$
Open section โ†’13.2

Mode of Grouped Data

Find the modal class and place the mode inside it with a formula that weighs the two neighbouring classes.

$\text{Mode}=l+\dfrac{f_1-f_0}{2f_1-f_0-f_2}h$
Open section โ†’13.3

Median of Grouped Data

Build the cumulative frequency column, find the median class, compute the median and test the relation among the three averages.

$\text{Median}=l+\dfrac{\frac n2-cf}{f}h$ ยท $3\,\text{Median}=\text{Mode}+2\,\text{Mean}$
Open section โ†’
โœ… Before you start
You need, from Class 9 and earlier: the mean, median and mode of a short ungrouped list, reading and drawing a frequency table and a bar graph or histogram, and fluent arithmetic with decimals and fractions. From this course you need to solve linear equations in one or two unknowns (Section 3.2), which you will use to find missing frequencies. Warm up: find the mean, median and mode of $2,\ 3,\ 3,\ 5,\ 7$ (answers: $4$, $3$, $3$).
โ† 12.2 Volume of a Combination of Solids13.1 Mean of Grouped Data โ†’