Learn on PengiThe Art of Problem Solving: Prealgebra (AMC 8)Chapter 13: Data and Statistics

Lesson 1: Basic Statistics

Grade 4 students in the AMC 8 Prealgebra course are introduced to basic statistics, learning how to calculate the mean (arithmetic average), median, and mode of a data set, as well as the range. The lesson covers key concepts such as finding the middle value of ordered lists, handling even-numbered data sets by averaging the two middle values, and understanding how the average relates to the sum of a list. This foundational lesson in Chapter 13 establishes the vocabulary and methods students will use to summarize and interpret numerical data.

Section 1

Measures of Center

Property

There are three such measures in common use: a) the arithmetic average of the data, b) the number that divides the data, when put in numerical order, into two pieces of the same size, c) the number that occurs the most often. In statistics these are respectively called: a) mean, b) median, c) mode. They each serve different purposes. A measure of center for a numerical data set summarizes all of its values with a single number.

Examples

For the data set {2, 5, 5, 6, 7}, the mean is 5, the median is 5, and the mode is 5. Here, all three measures of center are the same.
For the data set {3, 4, 4, 9}, the mean is 5, the median is 4 (the average of the two middle numbers), and the mode is 4.
For house prices {200000 dollars, 210000 dollars, 220000 dollars, 500000 dollars}, the median (215000 dollars) is a better measure of center than the mean (282500 dollars) because the mean is affected by the very high price.

Explanation

A measure of center is one number that represents a 'typical' value for a whole group of data. The mean, median, and mode are three different tools for finding this central value, each useful in different situations.

Section 2

The Mean: Fair Share and Balance Point

Property

The most common measure of center is the mean.
The mean is the arithmetic average, often referred to simply as “average.”
The procedure of computing the mean is to add up all the data values and then divide by the number of data values.
The significance of the mean is that it represents a fair share of the total.
For a data set of $N$ values, $a_1, a_2, …, a_N$ , the formula is:

= \frac{a1 + a2 + a3 + \cdots + aN}{N} $$
Another way to view the mean is as a balance point: the sum of the distances of the data points from the mean for those points below the mean is equal to the same sum for all the points above the mean.

Examples

A student's scores on five math tests are 85, 90, 75, 88, and 82. The mean score is calculated as $\frac{85+90+75+88+82}{5} = \frac{420}{5} = 84$ .
Four friends collected stamps: 30, 42, 25, and 35. To share them equally, they find the mean: $\frac{30+42+25+35}{4} = \frac{132}{4} = 33$ . Each friend gets 33 stamps.
The mean height of three plants is 15 cm. Two plants measure 12 cm and 18 cm. The height of the third plant, $h$ , is found by solving $\frac{12+18+h}{3} = 15$ , which gives $h = 15$ cm.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Measures of Center

Property

Examples

For the data set {2, 5, 5, 6, 7}, the mean is 5, the median is 5, and the mode is 5. Here, all three measures of center are the same.
For the data set {3, 4, 4, 9}, the mean is 5, the median is 4 (the average of the two middle numbers), and the mode is 4.
For house prices {200000 dollars, 210000 dollars, 220000 dollars, 500000 dollars}, the median (215000 dollars) is a better measure of center than the mean (282500 dollars) because the mean is affected by the very high price.

Explanation

Section 2

The Mean: Fair Share and Balance Point

Property

Examples

A student's scores on five math tests are 85, 90, 75, 88, and 82. The mean score is calculated as $\frac{85+90+75+88+82}{5} = \frac{420}{5} = 84$ .
Four friends collected stamps: 30, 42, 25, and 35. To share them equally, they find the mean: $\frac{30+42+25+35}{4} = \frac{132}{4} = 33$ . Each friend gets 33 stamps.
The mean height of three plants is 15 cm. Two plants measure 12 cm and 18 cm. The height of the third plant, $h$ , is found by solving $\frac{12+18+h}{3} = 15$ , which gives $h = 15$ cm.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Measures of Center

Property

Examples

For the data set {2, 5, 5, 6, 7}, the mean is 5, the median is 5, and the mode is 5. Here, all three measures of center are the same.
For the data set {3, 4, 4, 9}, the mean is 5, the median is 4 (the average of the two middle numbers), and the mode is 4.
For house prices {200000 dollars, 210000 dollars, 220000 dollars, 500000 dollars}, the median (215000 dollars) is a better measure of center than the mean (282500 dollars) because the mean is affected by the very high price.

Explanation

Section 2

The Mean: Fair Share and Balance Point

Property

Examples

A student's scores on five math tests are 85, 90, 75, 88, and 82. The mean score is calculated as $\frac{85+90+75+88+82}{5} = \frac{420}{5} = 84$ .
Four friends collected stamps: 30, 42, 25, and 35. To share them equally, they find the mean: $\frac{30+42+25+35}{4} = \frac{132}{4} = 33$ . Each friend gets 33 stamps.
The mean height of three plants is 15 cm. Two plants measure 12 cm and 18 cm. The height of the third plant, $h$ , is found by solving $\frac{12+18+h}{3} = 15$ , which gives $h = 15$ cm.