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Chapter 12: Statistics

We live in a world drowning in data: temperatures, populations, prices, exam scores, election votes. Statistics is the branch of mathematics that turns this raw data into understanding. This chapter is an introduction to the most basic tools of statistics: collecting data, organising it into frequency tables, displaying it with bar graphs and histograms, and summarising it with measures of central tendency , the mean, median, and mode.

The chapter is practical and visual. Most exam problems give you a list of data (test scores, daily temperatures, ages of students) and ask you to (1) organise it, (2) draw a graph, or (3) compute a summary statistic. The mathematics is easy , what matters is clarity of thought: what does the data represent, what is the right way to display it, and what does the summary statistic mean?

You will meet three kinds of data summaries: the mean (average), the median (middle value), and the mode (most frequent value). Each captures a different aspect of "typical" , and each has its strengths and weaknesses. The mean is sensitive to extreme values; the median is robust; the mode tells you what is most common. Understanding when to use each is the central insight of the chapter.

By the end you should be able to look at a list of data, organise it into a frequency distribution, draw a bar graph or histogram correctly, and compute the mean, median, and mode , and explain what each one tells you about the data.

What's inside

  • Collection and types of data , primary, secondary, ungrouped, grouped.
  • Frequency distribution , tables that summarise data.
  • Bar graphs and histograms , visual displays of data.
  • Mean, median, mode , three measures of central tendency.
  • Comparing measures , when each is most useful.

Key results / Formula card

ConceptStatement
RangeLargest value minus smallest value.
Class sizeWidth of an interval in grouped data.
FrequencyNumber of times a value (or class) occurs.
Mean (raw)xˉ=∑xin\bar{x} = \dfrac{\sum x_i}{n}
Mean (frequency)xˉ=∑fixi∑fi\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}
Median (ungrouped)Middle value when data is arranged in order; for nn even, average of the two middle values.
ModeValue with highest frequency.
Bar graphRectangles with equal-width bars; height = frequency.
HistogramLike a bar graph, but for continuous (grouped) data; no gaps between bars.

Memorise this card. Almost every question in this chapter is one of these definitions applied to a specific data set.

Sub-topics

5 pages
01
Collection and Types of Data
What counts as data, the difference between primary and secondary data, and how raw numbers get organised before any analysis.
02
Frequency Distribution Tables
Organise data into tables that show how often each value (or class) appears. Master both ungrouped and grouped frequency distributions.
03
Bar Graphs and Histograms
Visual displays of data. Bar graphs show categorical data with gaps between bars; histograms show grouped continuous data with no gaps.
04
Mean of Data
The arithmetic mean is the most common summary of data. Compute it for both ungrouped and grouped data, and understand its strengths and limitations.
05
Median and Mode
Two more measures of central tendency. The median is the middle value when data is ordered; the mode is the most frequent value. Each tells a different story than the mean.

Practice quiz

Answer the questions; explanations appear after each.

Quiz
Statistics : Mixed practice
10 questions · pick the best answer
Q1

The mean of 5, 7, 9, 11, 13 is:

Q2

The median of 4, 6, 8, 10, 12 is:

Q3

The mode of 2, 3, 3, 4, 5, 3, 6 is:

Q4

The range of 12, 15, 18, 9, 21 is:

Q5

Which is NOT a measure of central tendency?

Q6

If the mean of xx, x+2x+2, x+4x+4, x+6x+6, x+8x+8 is 11, then xx is:

Q7

The median of 7, 12, 4, 9, 15, 6 is:

Q8

In a bar graph, the height of each bar represents:

Q9

Class mark of the interval 20-30 is:

Q10

If ∑fi=20\sum f_i = 20 and ∑fixi=240\sum f_i x_i = 240, the mean is: