In this lesson, students examine multiple samples of the same population and learn what it means for a sample to be representative of the population. Students look at the structure of dot plots, attending to center, shape, and spread, to help them compare the samples and the population (MP7). Although the previous lesson pointed out the usefulness of using samples when working with large populations, the problems in this lesson use smaller populations so that students can compare each sample against the entire population.
- Calculate the mean or median of various samples, and compare them with the mean or median of the population.
- Comprehend that the term “representative” (in spoken and written language) refers to a sample with a distribution that closely resembles the population’s shape, center, and spread.
- Given dot plots, determine whether a sample is representative of the population, and explain (orally and in writing) the reasoning.
Let’s see what makes a good sample.
- I can determine whether a sample is representative of a population by considering the shape, center, and spread of each of them.
- I know that some samples may represent the population better than others.
- I remember that when a distribution is not symmetric, the median is a better estimate of a typical value than the mean.
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55 \div 5 = 11\).
mean absolute deviation (MAD)
The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11.
To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are.
\(4+2+1+2+3=12\) and \(12 \div 5 = 2.4\)
The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order.
For the data set 7, 9, 12, 13, 14, the median is 12.
For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14 \div 2 = 7\).
A population is a set of people or things that we want to study.
For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams.
A sample is representative of a population if its distribution resembles the population's distribution in center, shape, and spread.
For example, this dot plot represents a population.
This dot plot shows a sample that is representative of the population.
A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band.