Bell Ringer Create a stem-and-leaf display using the Super Bowl data from yesterday’s example. 47 23 30 29 27 21 31 22 38 46 37 56 59 52 36 65 39 61 69 43 75 44 61 45 31 46 31 66 50 37 47 44 47 56 55 53 39 41 37

Displaying and Summarizing Quantitative Data Chapter 4 Part 2 Displaying and Summarizing Quantitative Data

When describing data, we want to tell about 3 things: Its shape 2) Its center 3) Its spread

Shape Of a Distribution

Recall from Algebra class… mean, median, and mode. What does “mode” mean? The mode of a distribution is the value that occurs most often.

If the histogram has only one peak, it is called unimodal. The tallest bar represents the mode of the distribution in a histogram. If the histogram has only one peak, it is called unimodal.

Histograms that have two peaks are called bimodal while histograms that don’t appear to have any peaks are called uniform. bimodal uniform

Not Symmetric Symmetric histogram

If one tail of a histogram stretches out farther than the other, the histogram is skewed. Skewed Left Skewed Right

Look for anything unusual in the distribution, like gaps or outliers. An outlier stands away from the body of the distribution.

Center Of a Distribution

For a histogram that is symmetric or unimodal, the center is easy to find. It’s in the middle. It’s more difficult to find the center of the distribution if the histogram is skewed or bimodal.

Mean = Average Median = Middle Mode = Most The center can be described using either the mean, the median, or the mode. Mean = Average Median = Middle Mode = Most

Median or Mean? If the histogram is symmetric, use the mean. If the histogram is skewed or has outliers, the median is a better choice.

A few things to remember… If a distribution is symmetric, the mean and median will both be near the middle. If a distribution is skewed or has an outlier, the mean will be pulled in the direction of the skew or outlier.

If skewed right, the mean is pulled right (higher than the median) If skewed left, the mean is pulled left (lower than the median)

Spread Of a Distribution

Statistics is about variation. Are the values of the distribution clustered around the center or are they more spread out? Always report a measure of spread along with a measure of center when describing a distribution numerically.

Range range = max - min An outlier can make the range large and not representative of the data overall.

First, put the values in ascending order. Example: Find the median for the batch of values: 14.1, 3.2, 25.3, 2.8, -17.5, 1.5, 13.9, 45.8, 22.6 First, put the values in ascending order. -17.5, 1.5, 2.8, 3.2, 13.9, 14.1, 22.6, 25.3, 45.8 Median = 13.9

Quartiles divide the data into 4 equal sections. Interquartile Range Quartiles divide the data into 4 equal sections. It’s like finding the median, and then finding the median of each half.

Interquartile Range = 22.6 – 2.8 = 19.8 The difference between Q3 (upper quartile) and Q1 (lower quartile) is the interquartile range. -17.5, 1.5, 2.8, 3.2, 13.9, 14.1, 22.6, 25.3, 45.8 Q1 = 2.8 Q3 = 22.6 Interquartile Range = 22.6 – 2.8 = 19.8

The Interquartile Range contains the middle 50% of the data. Even if the distribution is skewed, the IQR almost always provides a reasonable summary of the spread.

5 Number Summary The 5-number summary includes the minimum, Q1, median, Q3, and maximum values of a data set. -17.5, 1.5, 2.8, 3.2, 13.9, 14.1, 22.6, 25.3, 45.8 Min = -17.5 Q1 = 2.8 Median = 13.9 Q3 = 22.6 Max = 45.8

Here are the first 42 Super Bowl total scores again. Find the median Find the quartiles Write a description based on the 5-number summary 47 23 30 29 27 21 31 22 38 46 37 56 59 52 36 65 39 61 69 43 75 44 61 45 31 46 31 66 50 37 47 44 47 56 55 53 39 41 37

TI Tips on page 65 explain how to find the 5-Number Summary on the calculator. Now use the calculator to check your answers for the Super Bowl scores 5-number summary.

Today’s Assignment Read Chapter 4 Homework: pg. 72 #9-11, 14, 18, 21