Section 11 - 7 Standard Deviation Day #2

Section 11 - 7 Standard Deviation Day #2

Solidifying our Understanding of Standard Deviations

Multiple Choice

What is the median?

• biggest- smallest
• average
• the middle #
• # that happens the most

Multiple Choice

What is the mean?

• # happening the most
• the number in the middle
• biggest-smallest
• the average

Multiple Choice

The mode is the number ________________________.

• you see most

• you see least

• you don't see

• you see first

Definitions

• Mean, Median and Mode are measures of a data set.

• Measure of Variation: DESCRIBES how the data in a data set are spread out.

• Standard Deviation and Variance are measures that show how much data values deviate from the mean.

Multiple Choice

What is the standard deviation for the data given:
5, 10, 7, 12, 0, 20, 15, 22, 8, 2

• 6.89
• 10.1
• 7.26
• 9

Multiple Choice

If the standard deviation of a data set is 4, what is the variance?

• 2
• 16
• 4

The Roster of the Bulls

• The chart at right shows the heights of the players on the Chicago Bulls.

• Create a FREQUENCY table of the values. Notice...they are already in ascending order...you're welcome!

Multiple Choice

Find the mean, median and mode of the heights of the Bulls. Using those values, is the data right-, left- tailed or normally distributed? If need - be, sketch a copy of the histogram. (bar chart)

• Right - Tailed

• Left - Tailed

• Normally Distributed

Normally Distributed

• Mean and Median are relatively close

• Why not right tailed? No mode. So mean and median being almost equal tells us normal distribution. So...let's look at the data spread.

Summarizing Spread

• Ignoring the percentages for now. We say if a data value lies within either one positive OR one negative sigma from the mean "it lies within one standard deviation of the mean."

• If it is +/- 2 sigmas then it is within two standard deviations of the mean.

• If it is +/- 3 sigmas, then it is within three standard deviations of the mean.

Multiple Choice

We say any value outside of 3 standard deviations from the mean is "statistically insignificant." It does not have any affect on our conclusions other than pulling the mean one direction or another. For the Bulls, are there any heights that are statistically insignificant? Meaning, more than 3 standard deviations from the mean?

• 75 inches

• 86 inches

• Both 75 and 86 inches

• All the data is within 3 standard deviations of the mean.

Multiple Choice

Within how many standard deviations do the heights of the Bulls players lie?

• One Standard Deviation

• Two Standard Deviations

• Three Standard Deviations

Reasoning

• As long as there is one data point within EITHER 3+ or 3- standard deviations we say the data lies within 3 standard deviations.

• There is no statistically insignificant data in this set.

Percentile vs. Z - Score

The box - and - whiskers plot summarizes specific percentiles, but all percentiles can be calculated. A curve and standard deviations summarize specific z - scores (+/- 1, 2, 3) but you can find specific z - scores within a data set using the formula at right.

Multiple Select

Which of the following z - scores are true for the indicated heights given the mean heights of the players is 79.12 inches and the standard deviation is 3.05. Select all that are correct

• 75 inches: z - score = -1.35

• 82 inches: z - score = 0.94

• 86 inches: z - score = -2.26

• 79 inches: z - score = 0.04

Hurricanes in Atlantic Oceans from 1992 to 2006

Enter the data into your LIST (Open up a Google sheet and pause here if you don't have a calculator to do this - we will talk through using a sheet) Otherwise - calculate the Mean, Standard Deviation and sketch a histogram of the information.

Multiple Select

Which of the following hold true about the number of hurricanes in the Atlantic Ocean for that 15 year period. Select all that apply.

• Mean: 7.267

• Standard Deviation: 3.214

• That data lies within 2 standards of the mean.

• Standard Deviation: 3.327

• 14 hurricanes is statistically insignificant data

Last Example

The chart represents 10 random weeks of the Dow Jones Industrial Average in 2008 and in 2020. Put this information into TWO different lists.

Open Ended

The standard deviation (spread) of the weeks in 2008 is greater than the standard deviation (spread) of the weeks in 2020. In your own words, describe what that means about the Dow Jones Industrial Average during those weeks. Be sure to mention within how many standard deviations all of the data points fall for each year.

Project Application

Use any remaining time to calculate the standard and indicated z - scores for your data.