Making Sense of Data (Part 1) Math 1 Erwin High School Fall 2014

Why Make Sense of Data?

Why Make Sense of Data? Every day, people are presented with data on television, on the internet, in newspapers, and in magazines.

Why Make Sense of Data? Examples: States release report cards for schools.

Why Make Sense of Data? Examples: States release report cards for schools. States release statistics on crime and unemployment.

Why Make Sense of Data? Examples: States release report cards for schools. States release statistics on crime and unemployment. Sports writers report batting averages and shooting percentages.

Why Make Sense of Data? Examples: States release report cards for schools. States release statistics on crime and unemployment. Sports writers report batting averages and shooting percentages. Many more . . .

Why Make Sense of Data? Making sense of data is important in every day life and in most jobs today.

Why Make Sense of Data? Making sense of data is important in every day life and in most jobs today. Usually, a first step in understanding data is to analyze a plot of the data.

Why Make Sense of Data? Let’s jump right in and look at some data together to begin discussing how to make sense of the numbers . . . On your paper, label it: Making Sense of Data 1.

Problem 1 As part of an effort to study the wild black bear population in Minnesota, the Department of Natural Resources had staff anesthetize and then measure the lengths of 143 black bears.

Problem 1 The following dot plots show the data that was collected in the study.

Problem 1 1. a. Why do you think the plots seen below are called dot plots?

Problem 1 1. b. What do you think the 2 dots above the 40 on the male bear graph represent?

Problem 1 1. c. Based on the plots, how many male bears were there? How many female bears?

Problem 1 1. d. Based on the plots, what was the length of the longest bear? Was it male or female?

Problem 1 1. e. How would you respond if asked how the two plots compare to each other?

Problem 1 1. f. Are the shapes of the graphs fundamentally alike or fundamentally different?

Problem 1 1. g. How would you describe the shapes?

Problem 1 1. h. Are there any lengths that seem to be outside of the overall pattern for either graph?

Problem 1 1. i. Compare the centers of the two graphs.

Problem 1 1. j. Compare the spreads of the two graphs.

Describing a Distribution

Describing a Distribution When describing a distribution it is important to include information about three things:

Describing a Distribution When describing a distribution it is important to include information about three things: Shape

Describing a Distribution When describing a distribution it is important to include information about three things: Shape Center

Describing a Distribution When describing a distribution it is important to include information about three things: Shape Center Spread

Describing Shape

Describing Shape Some distributions are approximately normal, or mound-shaped, where the distribution has one peak and tapers off on both sides.

Describing Shape Some distributions are approximately normal, or mound-shaped, where the distribution has one peak and tapers off on both sides.

Describing Shape Normal distributions are symmetric – the two halves look like mirror images of each other.

Describing Shape Some distributions have a tail stretching towards the larger values (on the right side of the graph).

Describing Shape These distributions are called skewed to the right.

Describing Shape The skewed direction (left or right) tells which side the tail of the graph is on.

Describing Shape Therefore, graphs that have a tail stretching towards the smaller values (on the left side) are . . .

Describing Shape Therefore, graphs that have a tail stretching towards the smaller values (on the left side) are . . .

Describing Shape Once again, the skewed direction is the side that has the tail.

Describing Shape If a graph has any values that seem unusually large or unusually small, these are referred to as outliers.

Describing Shape If a graph has any values that seem unusually large or unusually small, these are referred to as outliers. These should be mentioned when describing the shape of a graph.

Problem 2 2. a. What are the three possible shapes of distributions mentioned in the last section? (List the names and draw the corresponding shapes.)

Problem 2 2. b. What is an outlier?

Problem 2 2. c. How would you use the idea of skewness and outliers to describe the shape of the female bear distribution now?

Recap: Describing Shape We usually describe the shape of a distribution in one of three ways:

Recap: Describing Shape We usually describe the shape of a distribution in one of three ways: Approximately Normal Skewed to the Right Skewed to the Left

Recap: Describing Shape We usually describe the shape of a distribution in one of three ways: Approximately Normal Skewed to the Right Skewed to the Left Any unsually large or small values are called outliers.

Other Types of Plots

Other Types of Plots Before discussing measures of center, it is useful to note that there are other plots besides dot plots that we will see and use. Dot plots are useful when the data set is relatively small, but for large sets of data, histograms are useful.

Other Types of Plots A histogram has a number line across the bottom that is marked with a scale, just like a normal graph. The height of each bar represents the frequency which is the count of how many values are in that bar.

Other Types of Plots Any value that would fall right on a line between two bars is counted in the bar on the right. Example:

Describing Center

Describing Center When describing a distribution, we want to also be able to describe where the center of a distribution is located.

Describing Center When describing a distribution, we want to also be able to describe where the center of a distribution is located. To do so, we use one of two measurements:

Describing Center When describing a distribution, we want to also be able to describe where the center of a distribution is located. To do so, we use one of two measurements: Mean or Median

Describing Center Before discussing which one should be used and why, it is important for us to remember what the mean and median of a data set are.

Describing Center Mean – the average of a set of numbers.

Describing Center Mean – the average of a set of numbers. It can be calculated by adding all the values in the data set up and dividing the sum by the number of values in the set.

Describing Center Median – the middle value in a set of data when the numbers are ordered from least to greatest.

Describing Center Median – the middle value in a set of data when the numbers are ordered from least to greatest. If a data set has an odd number of values, it is the middle value.

Describing Center Median – the middle value in a set of data when the numbers are ordered from least to greatest. If a data set has an even number of values, the median is the average of the two middle values in the set.

Problem 3 Find the mean and median of the four data sets listed on the papers at the front of the room. For each set, list the name of the data set and the values of the mean and median (be sure to label which is which).

Describing Center When we look at a graph, the median is simply wherever the middle value is located.

Describing Center When we look at a graph, the median is simply wherever the middle value is located. In a normal distribution, the median is located near the peak of the graph.

Describing Center In a skewed distribution, the median is close to the peak of the graph, but shifted slightly towards the tail of the graph.

Describing Center Example:

Describing Center Example: The curve drawn doesn’t fit perfectly, but shows that the data is clearly skewed to the right. Example:

Describing Center Example: In the graph below, ticket prices from 20 concerts were used. Therefore, the median is the average of the 10th and 11th values when ordered from smallest to greatest. Example:

Describing Center Example: The 10th and 11th values both fall in the bar from $60 - $70, so the median occurs at this location in the plot. Median

Describing Center Example: The mean value of the ticket prices listed below comes out to around $75 per ticket. Median

Describing Center Example: This means that the mean is located slightly closer to the tail end of the graph than the median is. Mean

Describing Center One way to estimate the mean of a distribution is to imagine that the bars represent stacks of blocks and the mean is the location of the balance point.

Describing Center For a normal distribution, we already mentioned that the median is located at the peak (around 44 for the graph below). If the mean is located at the balance point, how does it compare to the location of the median?

Describing Center For normal distributions, the mean and median are both located right near the peak of the graph – they are close to being equal in value.

Describing Center So which one do we use? The mean or the median?

Describing Center So which one do we use? The mean or the median? The answer is . . .

Describing Center So which one do we use? The mean or the median? The answer is . . . it depends.

Describing Center So which one do we use? The mean or the median? The answer is . . . it depends. We have to look at the shape of the distribution first.

Describing Center For distributions that are approximately normal we use the mean as the best representation of the center (although the median should be close to the same value).

Describing Center For distributions that are approximately normal we use the mean as the best representation of the center (although the median should be close to the same value). For distributions that are skewed we use the median to represent the center.

Recap: Describing Center To describe the center of a distribution, we tell where the mean or median is located.

Recap: Describing Center To describe the center of a distribution, we tell where the mean or median is located. If the distribution is normal, we use the mean.

Recap: Describing Center To describe the center of a distribution, we tell where the mean or median is located. If the distribution is normal, we use the mean. If it is skewed, we use the median.

Problem 4 Take a sheet from the front with the two histograms. For each plot, label where you believe the mean and median would be located and circle which measure of center should be used to describe the center of the distribution based on the shape of the graph.

Describing Spread

Describing Spread We will discuss how to describe the spread in more detail in some of the next lessons.

Describing Spread We will discuss how to describe the spread in more detail in some of the next lessons. For now, you can use the range of a set of data to describe the spread.

Describing Spread The range of a set of data is the difference between the largest value (maximum) and smallest value (minimum).

range = maximum - minimum Describing Spread The range of a set of data is the difference between the largest value (maximum) and smallest value (minimum). range = maximum - minimum

Problem 5 For each of the four data sets you used in problem 3, determine the range of the data set.

Problem 6 Based on your understanding of shape, center and spread, describe the distribution for the male bear population graph from problem 1.

End of Part 1