AP Stat Day 1068 Days until AP Exam

Describing Data with NumbersBox and Whisker Plots

Variance and Standard Deviation

Today’s Objectives

Content: I can display data in a box and whisker plot and calculate outliers and standard deviation.

Language: I can work cooperatively to create a box and whisker plot using class data.

Calculating a Mean…

In statistics, the mean has a special symbol:

which is said “x-bar”

The formula for calculating the mean is:

Or, more simply:

x

n

xxxx n

...21

ixnx1

Calculating a Median

• Order the numbers from smallest to largest.

• If the number of observations, n, is odd then the median is the middle number.

• If the number of observations, n, is even, then the median is the mean of the two middle numbers.

Comparing the mean and the median

• The median is a resistant measure of center.

• This means that outliers and skewness do not effect the location of the median.

• The mean is a non-resistant measure of spread.

• This means that outliers and skewness strongly effect the mean.

Skewed Distributions

• In a perfectly symmetric distribution, the mean median and mode lie in the same place.

• In a right (positive) or left (negative) skewed distribution, the mean is pulled towards the tail. The mode remains at the highest point of the distribution.

Quartiles

• The median is the center point for a set of data. There is 50% of the data on each side of the median.

• A quartile contains 25% of a set of data.

• The first quartile, Q1, is the median of the bottom half of the data.

• The third quartile, Q3, is the median of the upper half of the data.

EXAMPLE100 92 45 88

91 36 72 63

56 83 97 77

65 92 81 51

78 86 92 21

Find the quartiles of the following data.Begin by ordering the numbers.

Find the median.

Find the median of the lower numbers.

Find the median of the upper numbers.

If there is an odd number of observations, leave out the median.

If there is an even number of observations, include all numbers in the list (the median is not in the list.

The IQR and Outliers

• The IQR is short for “Interquartile Range”

• To calculate IQR, IQR = Q3 - Q1

• Outliers are calculated using the IQR.

• The rule for outliers is that if a value is outside 1.5(IQR) then it is an outlier.

• So, if a value is more than Q3 + 1.5(IQR) or less than Q1 – 1.5(IQR) then it is an outlier.

Example100 92 45 88

91 36 72 63

56 83 97 77

65 92 81 51

78 86 92 21

Using the data from the last example, determine whether the data contains any outliers.

The 5 Number Summary

• The 5 number summary is the list of values used to create a box and whisker plot.

• Minimum, Q1, Median, Q3, Maximum

• We have already gathered all of this data, we just need to list it.

• Ready to use your calculator?

Drawing a Box and Whisker Plot

• Once you have your 5 number summary, drawing a box and whisker plot is easy.

• Important features: a number line, an x-axis title.

• The modified box plot

Activity

• On pg. 21, use the information that we gathered on class heights (back of pg 6) to make a box and whisker plot by hand.

• You may use the calculator for the 5 number summary, but you must make the box and whisker plot by hand. You must check to see if there are any outliers.

Box Plots in the Calculator

• We will use the height data already stored in the calculator from your activity.

Variance

• Variance is a number that shows the variation around the mean.

• The formula for calculating variance is:

• or, more simply:

€

s2 =(x1 − x)

2 + (x2 − x)2 + ...+ (xn − x)

2

n −1

€

s2 =1

n −1(x i − x)

2∑

Example

• Find the variance of the following list of numbers: 5, 7, 3, 12, 4

1)Find the mean

2)Find the difference of the mean and each observation

3)Square the differences

4)Sum the difference

5)Divide by n - 1

Standard Deviation

• The standard deviation is the square root of the variance.

• Find the standard deviation of the previous list of numbers.

€

s = s2

About Standard Deviation

• s measures spread about the mean

• s = 0 only when there is no spread (i.e. all observations have the same value)

• s, like x-bar, is not resistant! So, outliers or skewness makes s very large.

X-bar and s or 5 number summary?

• If your data is skewed or has strong outliers- use the 5 number summary!

• If your data is reasonably symmetric and free of outliers- use x-bar and s!

Summary (p 22)

• What numbers make up the 5 number summary?

• How does a modified box plot differ from a traditional box and whisker plot?

• If you know a variance, how can you find a standard deviation?

Prep Questions (p 23)

• Have you ever heard of a bell curve? What does it look like (words)? Sketch one.

• What does probability mean?

• What is density?