ALGEBRA UNIT 12- UNIVARIATE STATISTICS
Introduction to Statistics (Day 1)
Statistics is all about data. Without data to talk about or to analyze or to question, statistics
would not exist. There is a story to be uncovered behind all data - a story that has
characters, plots, and problems in the data. The questions or problems addressed by the
data and their story can be disappointing, exciting, or just plain ordinary! This unit is about
stories that begin with data.
Data: ___________________________________________________________________
Statistics: _______________________________________________________________
There are usually three steps in a statistical study:
1. The collection of ______________.
2. The organization of data into __________, _____________, and ____________.
3. The drawing of _________________ from an analysis of data.
Here are some ways to organize data:
Dot plots: ______________________________________________________________________________
1.
What does this graph tell us about who watched this television show?
2.
Can you make a conclusion about the type of show this data is about?
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Frequency Histograms: _________________________________________________________________
____________________________________________________________________________
Cumulative Frequency Histogram: ______________________________________________________
____________________________________________________________________________
3.
Is this a frequency histogram or a cumulative frequency histogram?
4.
What do you think this graph is telling us about the population of Kenya?
5.
Why might we want to study the data represented by this graph?
6.
Based on your previous work with histograms, would you describe this histogram as
representing a symmetrical or a skewed distribution? Explain your answer.
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Box Plots: ______________________________________________________________________________
______________________________________________________________________________
7.
What does the box plot tell us about the number of pets owned by the thirty
students at Binder City High School?
8.
Why might understanding the data behind this graph be important?
9.
What percent of the people in this experiment have 0 – 1 pets? 0 – 2? 0 – 5? 0 – 10?
**Remember**

Think of each graph as telling a story

Graphs of distributions are often the starting point in understanding the variability in
the data.

The types of graphs in the previous exercises will be analyzed in more detail in the
days that follow!
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MEASURES OF CENTER AND SPREAD (Day 2)

Mean ( x ): The _________ of the data values _______________ by how many values.

Median (Q2 or ____%): The _____________ value when data is in _____________!!!
o If there is no single middle number, take the average of the two middle
numbers.
Mode: The data value that ____________________________ occurs.
o No mode: If each number in a set of data occurs with the same frequency.
o Bimodal: When a set of data has two modes.

Quartile: When data is in numerical order, the number that separates data into _____ equal parts.


1st Quartile (Q1 or Lower Quartile or ____%): The ___________________ of the
_________________ half.
3rd Quartile (Q3 or Upper Quartile or ____%): The ___________________ of the
_________________ half.
Range: The __________________________ between the _____________________ and
______________________ data values.
Interquartile Range (IQR): The __________________________ between _____ and _____.
Data Set 1: Pet owners
Students from Binder City High School were randomly selected and asked, “How many
pets do you currently own?” The results are recorded below:
1.
Calculate the mean number of pets owned by the thirty students from Binder City
High School. Calculate the median number of pets owned by the thirty students.
Calculate the mode.
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Data Set 2:
Length of the east hallway at Binder City High School
Twenty students were selected to measure the length of the east hallway. Two marks were
made on the hallway’s floor, one at the front of the hallway and one at the end of the
hallway. Students were asked to use their meter sticks to determine the length between
the marks to the nearest tenth of a meter. The results are recorded below:
Now it’s your turn!! Create a dot plot for the data above.
1.
Why do you think that different students got different results when they measured
the same distance of the east hallway?
2.
Calculate the mean, median, and mode for this data.
Data Set 3:
Age of cars
Twenty-five car owners were asked the age of their cars in years:
Make a dot plot for the ages of cars:
1.
Describe the distribution of the age of cars. Find the median and mean. Would you
use the median or the mean to describe a typical age of a 25-year old car? Why??
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VARIABILITY/DEVIATION (Day 3)
Warm-up: Twenty-two students from the junior class and twenty-six students from the
senior class at River City High School participated in a walkathon to raise money for the
school’s band. Dot plots indicating the distances in miles students from each class walked
are shown below:
Estimate the mean for both graphs and put an “X” where you think it is.
What is the median distance for the seniors?
What is the median distance for the juniors?
Would you use the mean or median to describe the typical distance for the seniors? Why?
Would you use the mean or median to describe the typical distance for the juniors? Why?
Variability:
___________________________________________________________________________
Deviation:
___________________________________________________________________________
Life (Hours)
83
94
Deviation from the Mean
6
96
106
113
114
STANDARD DEVIATION (Sx): Shows how the data deviates from the ____________________.
A _________________ standard deviation indicates that the data tend to be very close to
the mean, while a ___________________ standard deviation indicates that the data are
spread out over a larger range of values.
TO FIND STANDARD DEVIATION BY HAND:
1.
2.
3.
4.
5.
Find the deviation from the mean for each data value.
Square each deviation value.
Add up all the squared values.
Divide the sum by the number one less than the total number of data values (n – 1).
Take this square root of the new value.
Life (Hours)
83
94
96
106
113
114
Deviation from the Mean
Squared Deviations
Add the numbers in the last row:
Divide this sum by (# of terms – 1):
this is the standard deviation (  x)!!
Take the square root:
Procedure to get Stat Measures using the Graphing Calculator:
1. Go to STAT, pick #1 (EDIT) - input stats into a list ( L1 )
2. Go to STAT  CALC, pick #1 (1-VAR STATS) {this gives you stats for 1 variable)
3. After 1-VAR STATS put L1 to indicate what list the data is in. (This may change)
Symbols to remember:
___________________, ___________________, ___________________, ___________________,
___________________
Using your calculator, find the following stat measures for: 77, 86, 84, 93, 90
a) Mean
b) Median
c) Standard Deviation
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THE FREQUENCY AND CUMULATIVE FREQUENCY HISTOGRAM (Day 4)
Frequency Histogram: a vertical ________ graph in which each interval is represented by
the width of the bar, the frequency of the interval is represented by the height of the bar.




No _____________ between bars.
A ______________ must be left at the beginning if the interval doesn’t start at ____.
A ______________ must be given to the _____________ and _____________ axes.
A _____________ must be given to the graph and each axis.
Example 1: A teacher graded a set of 32 test papers. The grades earned by the students
were as follows:
48, 85, 53, 42, 65, 62, 47, 95, 50, 82, 50, 58, 77, 93, 73, 55,
43, 66, 45, 44, 50, 49, 78, 70, 55, 95, 80, 78, 83, 81, 51, 60
Test Scores (Intervals)
41 – 50
51 - 60
61 - 70
71 - 80
81 – 90
91 - 100
Total Frequency:
Tally
Frequency (# of scores)
A. Complete the table
and construct a
frequency histogram
below.
B. Find the mean and median for the data. Then,
indicate where they fall on the histogram, using
arrows.
C. What is the shape of the distribution? Is it
symmetric, skewed to the right, or skewed to the
left?
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Skewed to the Left or
Symmetric or
Normal Distribution
Skewed to the Right or
Example 2: Label each graph as skewed right, skewed left, or symmetric.
a.
c.
b.
d.
Cumulative Frequency Histogram: A ________________ that displays the accumulation of
data from the ____________ interval of data to the _______________.

To find the cumulative frequency for each interval, ________ the frequency for
that interval to the frequencies of the ______________ intervals.
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Example 3: Construct a cumulative frequency histogram showing 8 intervals for these test
grades:
100, 93, 71, 74, 85, 56, 62, 68, 70, 100, 99, 85, 77,
85, 48, 51, 79, 25, 86, 93, 67, 88, 70, 100, 26
Test Scores
(Intervals)
B.
C.
D.
E.
Tally
Frequency Cumulative
(Number of Frequency
Scores)
How many students scored below a 61?
Which interval(s) have the greatest number of students?
What percent of these 25 grades were above 70?
Which 10-point interval contains the median?
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BOX (AND WHISKER) PLOTS (Day 5)
RECALL:
Median: When data is in numerical order, the number that separates data into _____ equal parts.
Quartile: When data is in numerical order, the number that separates data into _____ equal parts.
Procedure:
1. Arrange data from lowest to greatest
2. Find median (Second quartile/Q2)
3. Find First quartile (Lower quartile/Q1) - median of lower numbers
4. Find Third quartile (Upper quartile/Q3) - median of upper numbers
Interquartile Range: The _______________ between the __________ quartile (Q __ ) and
the ___________ quartile (Q __ ).
Example 1: The heights, in inches, of 20 students are shown in the following list. Find:
53, 60, 61, 63, 64, 65, 65, 65, 65, 66, 66, 67, 67, 68, 69, 70, 70, 71, 71, 73
a. The median
b. The first quartile
c. The third quartile
d. The interquartile range
Example 2: Using the following data, find the following with the help of your calculator.
8, 5, 12, 9, 6, 2, 14, 7, 10, 17, 11, 8, 14, 5, 6
c. The median
d. The first quartile
c. The third quartile
d. The interquartile range
BOX-AND-WHISKER PLOT :
5-number statistical summary: ________________, _________________, ________________,
__________________, and ___________________.
Procedure to get 5-number summary using the Graphing Calculator:
1. Go to STAT, pick #1 (EDIT) - input stats into a list
2. Go to STAT  CALC, pick #1 (1-VAR STATS) {this gives you stats for 1 variable)
3. After 1-VAR STATS put L1 to indicate what list the data is in. (This may change)
Symbols to remember:
___________________, ___________________, ___________________, ___________________,
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___________________
Procedure for drawing a Box and Whisker Plot
1. Determine 5 number summary using calc or by hand
2. Draw a scale with numbers from the minimum to the maximum value of a set of
data.
3. Place dots at the values of the five number statistical summary.
EXAMPLE 3:
4. Draw a box between the dots that represent the lower and upper quartiles, and a
A. Find the five statistical summary for the following set of data using your calculator:
vertical line in the box through the point that represents the median.
5. Add the whiskers by drawing a line segment from the minimum value to the box
and a line segment from the maximum value to the box.
Example 3: A data set consisting of the number of hours each of 40 students watched
television over the weekend has a minimum value of 3 hours, a Q1 value of 5 hours,
a median value of 6 hours, a Q3 value of 9 hours, and a maximum value of 12 hours.
Draw a box plot representing this data distribution.
2.
What is the interquartile range (IQR) for this distribution? What percent of the
students fall within this interval?
3.
Do you think the data distribution represented by the box plot is a skewed
distribution? Why or why not?
Example 4: For the following data, construct a box plot:
8, 5, 12, 9, 6, 2, 14, 7, 10, 17, 11, 8, 14, 5
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UNIVARIATE STATISTICS REVIEW (Day 6)
Example 1: Enter the following set of data into your calculator to construct a box and
whisker plot.
71, 68, 74, 80, 71, 30, 73, 67, 75, 74, 67, 68, 72, 69, 71
2.
What is the interquartile range (IQR) for this distribution? What percent of the
students fall within this interval?
3.
Do you think the data distribution represented by the box plot is a skewed
distribution? Why or why not?
Example 2: The accompanying histogram shows the heights of the students in Kyra’s
health class.
What is the total number of students in the class?
(1) 5
(2) 16
(3) 15
(4) 209
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Example 3: The following shows the grades of 10 students on three different quizzes.
a) On which quiz did the students tend to do the poorest?
b) Find the following for the data on quiz 1:
Mean =
Range =
Median =
Standard Deviation =
Example 4: Twenty students were surveyed about the number of days they played outside
in one week. The results of this survey are shown below. [6 pts.]
6, 5, 4, 5, 0, 7, 1, 5, 4, 4, 3, 2, 2, 3, 2, 4, 3, 4, 0, 7
Interval
0–1
2–3
4–5
6–7
Tally
Frequency
Cumulative Frequency
On the grid, create a cumulative frequency histogram based on the table you made.
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