Measures of Center vs Measures of Spread 1.2 Describing Data

Measures of Center Also known as measures of central tendency Determine the central point of a variable or the point around which all the data values are scattered. Main measures are: mean and median Mean = average. Can be described as the center of gravity, the point at which the whole group of data balances. Median = middle value. The point that divides all data values in half.

Measures of Spread Also known as “measures of variation” Describe how data values differ from each other and/or from their mean. Measures of spread are: range, interquartile range, and the standard deviation.

The range Range = largest value minus the smallest value Not a very reliable measurement because it depends only on the two extreme measurements and does not take into account the values of the remaining measurements.

Interquartile Range IQR IQR = Upper Quartile minus Lower Quartile Is the range of the middle 50% of the data. Is not affected by outliers

Standard Deviation Measures the distance between each data value and the mean. It is affected by outliers, therefore, when there are outliers the IQR is a better measurement. The larger the standard deviation, the wider the graph.

Standard Deviation Example The quiz scores of two students are listed below: Student A: 8, 9, 4, 8, 6, 8, 7, 9, 10, 5 Student B: 7, 8, 6, 6, 6, 6, 7, 5, 5, 7 Without calculating anything, whose scores have a greater spread? What student do you think has a greater standard deviation? Explain.

Calculating the standard deviation Calculate the mean Subtract the mean from each data value. Square each difference (deviation) Find the mean of the squared deviations. Take the square root of the mean of the squared deviations. NOTE: Make sure you create a table to organize your work =)

Review Questions A researcher takes a sample of 3000 flowers and measures their heights. Which is the best way to represent the data? A) bar chart B) stem plot C) histogram D) pie chart E) box and whisker plot

Review Questions 2. Which of the following data sets has the largest standard deviation? 1, 4, 7, 10, 13 1, 1, 3, 5, 5 1, 3, 5, 7, 9 1, 2, 3, 4, 5 1, 1, 1, 1, 1

Review Question 3. Which measure of center is best for skewed data? Mean Median Mode Standard deviation Interquartile range (IQR)

Review Questions Calculate the IQR for the following data 1 3 13 7 6 11 5 2 9 16 4 8 Answer 8

Review Question A teacher grades her students’ recent tests and computes the mean and the median: 84 and 86, respectively. While going over the test, she realized her key had an error. To correct for it, she added 4 points to everyone’s score. What are the new mean and median of the class? A) Mean: 84, Median 86 B) Mean: 88, Median 86 C) Mean 88, Median 90 D) Mean 84, Median 90 - answer is in next slide

Answer and explanation C If you add a positive constant to every data point in a sample, then the measures of center (mean and median) will increase by the value of that constant, respectively. In this problem, every student had 4 points added to their score, so the mean and median each increased by four points.

Review Question The summary statistics for the number of inches of snow in Vail, Colorado for 116 years is shown below. Determine whether there are outliers. N Mean Med StDev Min Max Q1 Q3 116 9.624 10.364 1.172 6.213 14.5 8.746 12.379 Answer: there are no outliers in this data set.

Assignment Core book page 570 PRACTICE Core book page 576 parts B and C Core book page 577 do all questions