Introduction Measures of center and variability can be used to describe a data set. The mean and median are two measures of center. The mean is the average value of the data. The median is the middle-most value in a data set. These measures are used to generalize data sets and identify common or expected values. Interquartile range and mean absolute deviation describe variability of the data set : Representing Data Sets

Introduction, continued Interquartile range is the difference between the third and first quartiles. The first quartile is the median of the lower half of the data set. The third quartile is the median of the upper half of the data set. The mean absolute deviation is the average absolute value of the difference between each data point and the mean. Measures of spread describe the variance of data values (how spread out they are), and identify the diversity of values in a data set. Measures of spread are used to help explain whether data values are very similar or very different : Representing Data Sets

Introduction, continued Data can be represented graphically using a number line. Graphs provide a visual representation of data; just by looking at a graph, you can quickly understand the spread and center of a data set. Dot plots and histograms show the frequency of a data value. In a dot plot, each data value is represented by a dot. The number of times a value is repeated corresponds to the number of dots above that value. In a histogram, the height of a rectangle above a value corresponds to the number of data values with that value. When looking at either a dot plot or histogram, it is easy to see both the most repeated data values and the spread of the data : Representing Data Sets

Introduction, continued If a data set is large, a histogram is easier to use because a single dot does not need to be drawn for each data value. A box plot shows the minimum, maximum, first quartile, median, and third quartile of numerical data. The middle 50% of the data is represented with a box. Lines on either side of the box extend to the minimum and maximum data values. A box plot shows the range of data in a data set, and measures of center can be easily seen on a box plot. Box plots can be used to compare expected values of multiple data sets : Representing Data Sets

Key Concepts Numerical data can be represented graphically on the real number line. Dot plots and histograms show the frequency of each data value in a data set. Each data value in a data set is represented by a dot over that value in a dot plot. In a histogram, a rectangle is drawn above each value in a data set. The height of each rectangle corresponds to the number of data points with that value : Representing Data Sets

Key Concepts, continued A histogram can show the frequency of a range of values. The minimum, maximum, first quartile, median, and third quartile of a data set must be calculated before creating a box plot. In a box plot, a rectangle is drawn starting at the first quartile and ending at the third quartile. The rectangle shows the middle 50% of the data set. The median is represented in the rectangle by a line. Whiskers are drawn from the rectangle to the minimum and maximum data values : Representing Data Sets

Key Concepts, continued A box plot shows more information about the expected value of a data set than a dot plot or histogram. A dot plot or histogram provides information about the size of a data set, which cannot be seen in a box plot : Representing Data Sets

Common Errors/Misconceptions confusing the mean and median using the mean in a box plot instead of the median incorrectly setting up the number line before creating a dot plot or histogram forgetting to include data values on a dot plot or histogram because they are not labeled on the number line : Representing Data Sets

Guided Practice Example 1 A pharmacy records the number of customers each hour that the pharmacy is open. The staff is using the information to determine how many people need to be working at the pharmacy at each time of day. The number of customers is in the table on the next slide. Use the table to create a histogram to help the pharmacy staff understand how many customers are in the pharmacy at each time of day : Representing Data Sets

Guided Practice: Example 1, continued : Representing Data Sets Time frameNumber of customers 8:00 A. M.–9:00 A. M.2 9:00 A. M.–10:00 A. M.0 10:00 A. M.–11:00 A. M.8 11:00 A. M.–12:00 P. M.14 12:00 P. M.–1:00 P. M.23 1:00 P. M.–2:00 P. M.12 2:00 P. M.–3:00 P. M.7 3:00 P. M.–4:00 P. M.3 4:00 P. M.–5:00 P. M.5

Guided Practice: Example 1, continued 1.Draw a number line on an x-axis that corresponds to the range of the data. The x-axis for this data will show the times the customers were counted. The number line for the pharmacy must include the times from 8:00 A. M. until 5:00 P. M. If using a number line that counts by twos, extend the number line to 6:00 P. M : Representing Data Sets

Guided Practice: Example 1, continued 2.Draw a y-axis that corresponds to the least and greatest number of times a data value is repeated. The y-axis should be to the left of the labeled x-axis. The number of customers arriving in each time frame ranges from 0 customers to 23 customers. The y-axis needs to show values from 0 to 23. If using a number line that counts by twos, extend the number line to : Representing Data Sets

Guided Practice: Example 1, continued : Representing Data Sets

Guided Practice: Example 1, continued 3.Create a rectangle at each value showing the number of data points at each data value. The rectangles will each span an hour, and will show the number of customers in that hour. There will be no rectangle from 9:00 A. M. to 10:00 A. M., because there were no customers at that hour : Representing Data Sets

Guided Practice: Example 1, continued : Representing Data Sets ✔

Guided Practice: Example 1, continued : Representing Data Sets

Guided Practice Example 3 The website Rate My Phone conducts reviews of smartphones. One aspect of the phones that is tested is battery life. The minutes of battery life for the newest 25 phones is recorded in the tables on the next two slides. Draw a box plot to represent the data : Representing Data Sets

Guided Practice: Example 3, continued : Representing Data Sets Smartphone Minutes of battery life Smartphone Minutes of battery life A380K360 B530L550 C350M370 D390N470 E520O280 F520P300 G430Q440 H330R490 I550S530 J290T340

Guided Practice: Example 3, continued : Representing Data Sets Smartphone Minutes of battery life Smartphone Minutes of battery life U250X520 V260Y320 W730

Guided Practice: Example 3, continued 1.Order the data from least to greatest. Note the minimum and maximum data values. The minimum data value is 250, and the maximum data value is : Representing Data Sets

Guided Practice: Example 3, continued : Representing Data Sets Smartphone Minutes of battery life Smartphone Minutes of battery life U250M370 V260A380 O280D390 J290G430 P300Q440 Y320N470 H330R490 T340X510 C350E520 K360F520

Guided Practice: Example 3, continued : Representing Data Sets Smartphone Minutes of battery life Smartphone Minutes of battery life B530L550 S530W730 I550

Guided Practice: Example 3, continued 2.Find the median of the data. The median is the middle-most data value. There are an odd number of data values, so the median is the 13th data value, : Representing Data Sets

Guided Practice: Example 3, continued 3.Find the first quartile of the data. The first quartile is the middle-most value of the lower half of the data. There are 12 data values in the lower half of the data, so the first quartile is the average of the sixth and seventh data values (320 and 330) : Representing Data Sets

Guided Practice: Example 3, continued 4.Find the third quartile of the data. The third quartile is the middle-most value of the upper half of the data. There are 12 data values in the upper half of the data, so the third quartile is the average of the 19th and 20th data values (520 and 520) : Representing Data Sets

Guided Practice: Example 3, continued 5.Draw a number line that includes the minimum and maximum data values. The minimum data value is 250, and the maximum data value is 730. If counting by 50s, extend the number line to : Representing Data Sets

Guided Practice: Example 3, continued 6.Draw a box, beginning at the first quartile (325) and ending at the third quartile (520) : Representing Data Sets

Guided Practice: Example 3, continued 7.Draw a line in the box at the median (390) : Representing Data Sets

Guided Practice: Example 3, continued 8.Draw a point at the minimum and maximum data values (250 and 730) : Representing Data Sets

Guided Practice: Example 3, continued 9.Connect the minimum and maximum data values to the box : Representing Data Sets ✔

Guided Practice: Example 3, continued : Representing Data Sets