Warm Up 1. Find the mean, median, mode, range, and outliers of the

following data.11, 7, 2, 7, 6, 12, 9, 10, 8, 6, 4, 8, 8, 7, 4, 7, 8, 8, 6, 5, 9

2. How does an outlier affect the data set?

3. Draw an example of a histogram that is skewed, and draw an example of one that is symmetrical.

4. Of 35 judges’ scores awarded during a gymnastics meet, 28 are less than or equal to 7.5. What is the percentile rank of a score of 7.5?

4 5 6 7 8 9 10 11 12

MedianLower

QuartileUpper

QuartileLowest Value

Highest Value

BoxWhiskerWhisker

130 140 150 160 170 180 190

Boys

Girlscm

Box and Whisker Diagrams.

Box plots are useful for comparing two or more sets of data like that shown below for heights of boys and girls in a class.

Anatomy of a Box and Whisker Diagram.

Box Plots

2. The boys are taller on average.

Question: Gemma recorded the heights in cm of girls in the same class and constructed a box plot from the data. The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers.

Drawing a Box Plot.

130 140 150 160 170 180 190

Boys

Girls

cm

1. The girls are taller on average.

3. The girls show less variability in height.

4. The boys show less variability in height.

5. The smallest person is a girl.

6. The tallest person is a boy.

Comparing Box and Whisker Plot

• We can see that in general the scores in the first approval poll are lower and not as wide spread as those of the second approval poll.

1 2 3 4 5 6 7 8

Approval Poll One

Approval Poll Two

1. Which has the highest death rate?

2. How many years of smoking has the least amount of variability?

3. Which one has the smallest median?

Which type of house has a higher average cost?Which house has the highest price?Which house has the lowest price? Which has the highest median?Which type has a larger spread, or variability in pricing?

STANDARD DEVIATION

Standard Deviation: tells us how far, on average, our data points are from the mean.

EX: Given the following data set: 2, 4, 4, 4, 5, 5, 7, 9The mean is 5, and the standard deviation is 2. That means that on average, the data points are 2 units away from the mean of 5.

Depart from what’s normal

Calculate Standard Deviation

Calculate Standard DeviationLuckily, we can also find it in our calculator by:1. Going to STAT then Edit2. Typing our data set into L1

3. Pressing STAT then CALC and 1-Var Stats4. You will see a lot of numbers. The standard deviation

looks like this: σx =

**If it is a number with a lot of decimals, you can round to the hundredths place (2 digits after the decimal)

Other Fun Things From the Calc.•x= minX=

•∑x= maxX=

•n = Med=

Examples:•Find the standard deviation of:

-13.3, 19.4, -33.1, -6.4, -8.2, 2.9, 4.3, -21, 1.7, and -21

Examples (cont.):•Find the standard deviation of: 1. 96, 53, 53, 48, 58, 94, and 33

2. -44.4, 5.2, 3.2, 14.3, -33.2, 10.7, 8.4, 8.2, 14.3, and 4.2

Using Standard Deviation• If you are given mean and standard deviation and told to

find a particular value...

•Start with the mean

•Add or subtract the standard deviation from the mean until you get what you need.

Using Standard Deviation•Mrs. Inscoe graded all of the Algebra I quizzes and found

that they had a mean of 80 and a standard deviation of 4. She tells you that you earned a score of 1 standard deviation above the mean, but that another student earned a score 1 standard deviation below the mean. What are your two scores?

Using Standard Deviation•You look up your favorite basketball player’s scoring

average and find that his mean points scored per game is 22 and his standard deviation is 6. If he scored 2 standard deviations below the mean in his game against the Bobcats, how many points did he score in that game? What about if he had scored 1 standard deviation above the mean?

Using Standard Deviation•Drivers in Charlotte commute on average 8 miles to get to

work, with a standard deviation of 2 miles. If you have to commute 3 standard deviations above the mean, how far do you have to go to commute to work?

Shapes of Data•Uniform- roughly the same height

throughout

•Symmetric- you could draw a vertical line that divides the graph into two parts that are close to mirror images

•Skewed- 1 peak, not in the center (left, and right)

•Cumulative Frequency Table- a frequency table that has an additional column that calculates a total for the data at each interval

Even More Vocabulary

Height Data

Height Interval FrequencyCumulative Frequency

4’7” – 5’5’1” – 5’6”5’7” – 6’

6’1” – 6’6”

You Try!•Complete the Frequency Table

•Histogram - a graph that shows data from a frequency table

▫Height shows frequency of the interval▫No gaps between bars▫Bars have equal widths

Still More Vocabulary

Height Data

Box & Whisker Plots

•Q1 is the lower quartile & represents the values in the lower 4th (bottom 25%) of the data

•Q3 is the upper quartile & represents the values in the upper 4th (top 25%) of the data

•Q2 is the median•The box from Q1 to Q3 is the middle 50% of the data

known as the interquartile range

Five Number Summary

Min valueQ1

Q2

Q3

Max Value

Box & Whisker Plots•Describe the data represented by the box and whisker plot.

Include the extreme values and quartiles.1.

2.

Examples•Find the minimum, first quartile, median, third quartile, and

maximum of each data set.1. 18, 14, 15.8, 9, 12, 16, 20, 16, 13, 15

2. 125, 80, 140, 135, 126, 140, 350, 75

Outliers• If the data set has outliers, they are represented by bullets,

do not include them in the whiskers. •Remember these outliers skew the mean and standard

deviation!▫Outlier is any element of a set of data that is at least 1.5

interquartile ranges less than the lower quartile (Q1) or greater than the upper quartile (Q3)

Q1 – 1.5IQR Q3 + 1.5IQR

Outliers•Example: Identify any outliers.

12, 12, 17, 23, 23, 23, 24, 24, 25, 26, 26, 28, 28, 29, 30, 31, 34, 36, 40, 46

Making a Box & Whisker Plot•Daily attendance: 29, 24, 28, 32, 30, 31, 26, 33, 14

In the calculator, enter data in L1

and choose the box and whisker icon in the StatPlot menu.

Box & Whisker Plot Shapes

Percentiles•Percentiles separate data sets into 100 equal parts. The

percentile rank of a data value is the percentage of data values that are less than or equal to that value.

•Examples: Of 25 test scores, eight are less than or equal to 75 and seven scores are between 75 and 85.▫What is the percentile rank of a test score of 75?

▫What is the percentile rank of a test score of 85?

Examples (cont.)2. Of 10 bowling scores, six are less than or equal to 120.

What is the percentile rank of a bowling score of 120?

3. Of 35 judges’ scores awarded during a gymnastics meet, 28 are less than or equal to 7.5. What is the percentile rank of a score of 7.5?