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?
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.
Lower
Lowest
Quartile
Value
Whisker
4
5
Median
Upper
Quartile
Whisker
Box
6
7
Highest
Value
8
9
10
11
12
Boys
130
140
150
160
170
180
cm
Girls
Box
Plots
190
Drawing a Box Plot.
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.
Boys
130
140
150
160
170
180
cm
Girls
1. The girls are taller on average.
2. The boys are taller on average.
3. The girls show less variability in height.
5. The smallest person is a girl.
4. The boys show less variability in height.
6. The tallest person is a boy.
190
Comparing Box and Whisker Plot
1
2
3
4
5
6
7
8
Approval Poll One
Approval Poll Two
• 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. 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
Depart from
what’s normal
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, 9
The 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.
Calculate Standard Deviation
Calculate Standard Deviation
Luckily, we can also find it in our calculator by:
1. Going to STAT then Edit
2. Typing our data set into L1
3. Pressing STAT then CALC and 1-Var Stats
4. 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
Height Data
Height Interval
4’7” – 5’
5’1” – 5’6”
5’7” – 6’
6’1” – 6’6”
Frequency
Cumulative
Frequency
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
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 value
Q1
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?