Data, frequency tables and graphs | Probability and statistics

Describing data

Statistics is the science of collecting, organizing and interpreting data. A few key words:

the population is the entire group you want to know about (all students at a school);
a sample is the part of the population you actually collect data from;
a variable is the characteristic you record.

A variable is categorical if its values are labels (favorite sport, eye color) and quantitative if its values are numbers you can do arithmetic with (height, number of pets).

Frequency tables

Twenty-five students were asked their favorite sport.

Sport	Frequency	Relative frequency	Percent
soccer	$$10$$	$$0.40$$	$40\%$
basketball	$$7$$	$$0.28$$	$28\%$
baseball	$$5$$	$$0.20$$	$20\%$
other	$$3$$	$$0.12$$	$12\%$
total	$$25$$	$$1.00$$	$100\%$

Relative frequencies always add up to $$1$$ : a quick way to check your table.

For quantitative data you can also find the cumulative frequency: the number of observations at or below each value.

Choosing a graph

Graph	Best for
bar graph	comparing categories
pie chart	showing parts of a whole
histogram	quantitative data grouped into intervals (bars touch)
dot plot	small quantitative data sets, one dot per value
stem-and-leaf plot	small data sets, keeping every value visible
line graph	change over time

In a pie chart, each slice's angle is the relative frequency times $$360°$$ .

ExampleA stem-and-leaf plot

Quiz scores $$62, 68, 71, 75, 75, 79, 84, 88, 93$$ :

Stem	Leaves
$$6$$	$2 \; 8$
$$7$$	$1 \; 5 \; 5 \; 9$
$$8$$	$4 \; 8$
$$9$$	$$3$$

The stem is the tens digit and each leaf is a ones digit. You can read off every original score, and the shape shows that most scores are in the $$70$$ s.

Practice

Quick check

Which variable is categorical?

Quick check

Which graph is best for showing how a city's population changed from 1950 to 2020?

Sport	Frequency	Relative frequency	Percent
soccer	$\(10\)$	$\(0.40\)$	$40\%$
basketball	$\(7\)$	$\(0.28\)$	$28\%$
baseball	$\(5\)$	$\(0.20\)$	$20\%$
other	$\(3\)$	$\(0.12\)$	$12\%$
total	$\(25\)$	$\(1.00\)$	$100\%$

Stem	Leaves
$\(6\)$	$2 \; 8$
$\(7\)$	$1 \; 5 \; 5 \; 9$
$\(8\)$	$4 \; 8$
$\(9\)$	$\(3\)$