+ Chapter 1: Exploring Data Section 1.2 Displaying Quantitative Data with Graphs – Histograms The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

+ If the directions read “compare the distribution of SAT scores between School 1 and School 2,” what is really being asked?

+ 1)Divide the range of data into classes of equal width. 2)Find the count (frequency) or percent (relative frequency) of individuals in each class. 3)Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals. Displaying Quantitative Data Histograms Quantitative variables often take many values. A graph of thedistribution may be clearer if nearby values are groupedtogether. The most common graph of the distribution of onequantitative variable is a histogram. How to Make a Histogram

+ Note that the axes are labeled! The bars have equal width!!! The height of each bar tells how many students fall into that class. The range of values on the x-axis is called a class.

+ Making a Histogram The table on page 35 presents data on the percent ofresidents from each state who were born outside of the U.S. Displaying Quantitative Data Example, page 35 Frequency Table ClassCount 0 to <520 5 to < to < to < to < to <301 Total50 Percent of foreign-born residents Number of States

+ SOCS for a Histogram…Have fun!

+ 1)Don’t confuse histograms and bar graphs. Why??? 2)Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations. 3)Make sure you label your classes appropriately (equally) Displaying Quantitative Data Using Histograms Wisely Here are several cautions based on common mistakesstudents make when using histograms. Cautions

+ Describing Quantitative Data Measuring Center: The Mean The most common measure of center is the ordinaryarithmetic average, or mean. Definition: To find the mean (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is: In mathematics, the capital Greek letter Σis short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation:

+ Describing Quantitative Data Measuring Center: The Median Another common measure of center is the median. Definition: The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1)Arrange all observations from smallest to largest. 2)If the number of observations n is odd, the median M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list.

+ Describing Quantitative Data Measuring Center Use the data below to calculate the mean and median of thecommuting times (in minutes) of 20 randomly selected NewYork workers. Example, page Key: 4|5 represents a New York worker who reported a 45- minute travel time to work. Wait, why are these values different???