4.3 Lesson

1. 4.3 Lesson

2. Median • Median (M) • Midpoint of a distribution which has been ordered from smallest to largest. Half of the numbers are below the midpoint, half are above. • If there is an odd number of entries in list, the median (M) is the middle number after list has been organized. • If there is an even number of entries in list, the median (M) is the average of the two middle numbers after list has been organized.

3. Quartiles • The first quartile (Q1) is the median of the first half of the data. • The third quartile (Q3) is the median of the second half of the data.

4. Five Number Summary • Consists of the minimum, 1st quartile, Median, 3rd quartile, and maximum.

5. Boxplot • A graph of the five number summary. • A central box spans the quartiles. • A line in the box marks the median. • Lines extend from the box to the largest and smallest numbers that are not outliers. • IQR = Inter Quartile Range = Q3 - Q1

6. Boxplot with no Outlier

7. Boxplot with an Outlier

8. Calculating an Outlier • Multiply 1.5 by the value of (Q3-Q1). • If the distance from Q1 to the suspected number is larger than the value of the above expression = an outlier. • If the distance from Q1 to the suspected number is smaller than the value of the above expression = not an outlier. • Works the same on the other end, but uses Q3 instead of Q1

9. For Example… • You have the following data… • 57, 85, 96, 97, 99, 106, 110, 113, 113 • The min is 57, max is 113 • Q1 is 90.5, Q3 is 111.5 • Median is 99 • 1.5(111.5-90.5) = 1.5(21) = 31.5 • The distance between 90.5-57 = 33.5 • Because the distance is larger, 57 is an outlier.

10. Mean • Is the symbol for the mean (pronounced x-bar). • It’s the average of all values (take the sum of all observations and divide by the number of observations.

11. Standard Deviation (labeled “s”) • Describes the average distance numbers are from their mean. • It is found by doing the following: • 1. Calculating the deviations (difference each number is from the average) (x-x bar). • 2. Square each value from #1 and add their values together. This is represented with a symbol • 3. Take the value from #2 and divide it by the number of values – 1 (n-1) (this represents a value called the variance). • 4. Take the square root of the value from #3. This represents the standard deviation.

12. Deviations Example

13. Example 4.19 (page 257-258) • Data: 1792, 1666, 1362, 1614, 1460, 1867, 1439 • Step 1: Find the mean • 1600 calories • Step 2: Write a list of all the observations • 1792, 1666, 1362, 1614, 1460, 1867, 1439 • Step 3: Find all the deviations • 192, 66, -238, 14, -140, 267, -161 • Step 4: Find all the squared deviations • 36864, 4356, 56644, 196, 19600, 71289, 25921

14. Example, page 2 • Step 5: Add up all the squared deviations • 214,870 • Step 6: To find the variance, take the answer from step 5 and divide it by n-1 (or 6) • 35,811.67 • Step 7: To find the standard deviation, take the answer from step 6 and square root it • 189.24 calories

15. Properties of Standard Deviation • s measures the spread about the mean. • Use s to describe the spread only when you use x-bar to describe the center. • s = 0 when there is no spread. Otherwise s > 0, which will get larger the more spread out the values are.

16. Standard Deviation • In the earlier example, the standard deviation was 189…seems like a large number, but when you take into account that the observed values ranged from 1362-1867, it’s not as intimidating a number (still large, just not as large as previously thought).

17. Choosing a Summary • Standard Deviation is strongly affected by outliers and by a skewed graph. Quartiles and the median are less affected. • It is a better idea to use the 5 number summary to describe a set of data if there are outliers or a skewed distribution. • It is OK to use standard deviation and mean to describe a distribution if the data set produces a symmetric distribution and is free of outliers.

18. Homework • Page 250, #4.42, 4.43 • Page 254-256, #4.46, 4.47, 4.49