Transcript Common Core Math 1 Statistics! WELCOME!!!

Common Core Math 1 Statistics
WELCOME!!!
Common Core State Standards
Reason quantitatively and use units to solve problems. Summarize,
represent, and interpret data on a single count or measurement variable.
N-Q.1 Use units as a way to understand problems and to guide the solution
of multi-step problems; choose and interpret units consistently in formulas;
choose and interpret the scale and origin in graphs and data displays.
N-Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N-Q.3 Choose a level of accuracy appropriate to the limitations on
measurement when reporting quantities.
S-ID.1 Represent data with plots on the real number line (dot plots,
histograms, and box plots).
S-ID.2 Use statistics appropriate to the shape of the data distribution to
compare center (median, mean) and spread (interquartile range, standard
deviation) of two or more different data sets.
S-ID.3 Interpret differences in shape, center, and spread in the context of the
data sets, accounting for possible effects of extreme data points (outliers).
Common Core math 1 suggested pacing
Day
1
CC1A (10 days)
Introduction to Statistics,
Categorical vs. Quantitative Data
2
Frequency Tables and Histograms
3
4
5
6
7
8
9
10
CC1 (5 days)
Intro to Statistics, Categorical vs. Quantitative (clickers);
begin Histograms
Frequency Tables, Histograms, Dot Plots (do activity with
paper clips); use graphing calculator
Histograms vs. Dot Plots; Using the Measures of Center
Graphing Calculator to make a
Histogram
Measures of Center (focus on
Measures of Spread and Outliers
Median)
Measures of Center (focus on
Test (have projects due Tuesday)
Mean)
Measures of Spread (focus on
Standard Deviation)
Boxplots. Outliers; discussion on
data
Activity: Comparing Data Sets
Review Using Data
Test; Projects Due
How do we use statistics?
TO DESCRIBE
(Data Analysis)
TO PREDICT
(Statistical
Inference)
Definitions
 Data: A collection of information in context.
 Population: A set of individuals that we wish
to describe and/or make predictions about.
 Individual: Member of a population.
 Variable: Characteristic recorded about each
individual in a data set.
Types of Variables
 Categorical Variable: A variable that records
qualities or characteristics of an individual,
such as gender or eye color.
 Quantitative Variable: A variable that
measures a characteristic of an individual,
such as height, weight, or age.
 Makes sense to average
 In this unit, we will focus on quantitative data.
Categorical or Quantitative Data?








Birth month
Number of siblings
Height in inches
Average amount of time (in minutes) of your ride
to school.
Number of pets
Year & model of the car you drive
Age of your youngest parent
Predicted letter grade of your first Math 1 test.
Categorical - GRAPHIcally
BAR GRAPH
CIRCLE GRAPH
Quantitative - Graphically
 Dotplot
 Histogram
 Boxplot
Let’s collect some data
Paper clip activity
Materials: paperclips, post it notes, (possibly two colors), whiteboard with 2
separate drawn and labeled axes
Directions:
• Give students first color post it note. Ask students to predict how many paperclips
they think they can link in 1 minute – have them put this guess on a post-it note
• Have them come to the board and put it on one of the axes (to form a dot plot)
• Give each student a set of paper clips – tell them when to start linking the paper clips
(time a minute) and tell them to stop – watch for cheaters!
• Have students count the number of paper clips (pass out other color post it note),
put that number of the post it note, and put on the other axes on the board (to form
a 2nd dot plot) Prize for the most linked?
Discussion:
• Compare and contrast the two graphs. Is this categorical or quantitative? This is
where you can make a frequency table and/or a histogram with the data and note
the differences. Could you make a bar graph or a pie graph? (no since you use those
for categorical data)
When Describing quantitative data
from a graph
Shape
Outliers
Center
Spread
Symmetric/Mound Shaped
Skewed Left
Any
unusual
values, outside the
Estimate
thelow
median
(extreme
values)
normal
trend
of data
The
middle
value,
same
Skewed
Right
number
ofhigh
values
(extreme
values)on the left
and right side
Uniform
Estimate the mean
IsThe
the data
spread point”
out? How
“balancing
or far is
theaverage
range? (max – min) Do you
think this would have a large or
small standard deviation?
Quantitative - Graphs
We suggest having the students create once by hand and the rest of
the time by calculator. It is more essential the student understands
how to interpret and compare/contrast graphs.
Histograms
Boxplots
 Commonly used over the boxplot
 How many bins are sufficient? 5-10
is standard
 If a data point falls on the tick
mark value, does it go in the bin on
the left or right? On the right
 The calculator does not always
choose an appropriate bin width –
to change, you can change your x
scaling on the calculator under
WINDOW
 Understand the TRACE feature
 Harder to manipulate the data
than the histogram
 Know the difference between the
two box plot options in the
calculator – one shows outliers
with a dot, the other does not
show outliers
 Five number summary
 Understand how to graph a side by
side box plot – put the other data
in L2 and turn the 2nd STATPLOT On
 25% of the data falls within each
quartile
Describing Numerically
Categorical Data
Quantitative Data
 Counts
 Proportions
 Percents
 Center
 Mean
 Median
 Variability/Spread
 IQR
 MAD
 Standard Deviation
 Measures of Relative
Standings (ex: z-scores) – we
do not go into this in CCM1
Describing quantitative Numerically
CENTER: mean vs. median
Mean
Median
Use the mean to describe the middle of
set of data that does not have an outlier
Use the median to describe the
middle of a set of data that does
have an outlier
Advantages
 Extreme values (outliers) do
not affect the median as
strongly as they do the mean.
 Useful when comparing sets of
data.
Disadvantages
 Not as popular as mean.
μ (with entire population) or x (with a
sample of a population)
Advantages
 Most popular measure in the fields
such as business, engineering, and
computer science.
 Useful when comparing sets of data
Disadvantages
 Affected by extreme values (outliers)
x
sum of data values
number of data values
x
x
n
http://regentsprep.org/REgents/math/ALGEBRA/AD2/measure.htm
Describing quantitative Numerically
CENTER: mean vs. median
EXAMPLE : The football team had a fundraiser and the coach
gladly announced that the average (or the mean) of sales for
each football player was approximately $90. Now look at
the data of the football players’ fundraiser sales below.
When you check the coach’s math, he was telling the truth –
the average (or the mean) is $87.50, but why is this not an
accurate representation of the data?
 Make a histogram by hand or by calculator. Sketch the
graph below.
 Circle the outlier.
 On the histogram, label the mean ($87.50) with a line on the
graph.
 Then calculate the median and draw where the median is
located on the graph with a line.
 Should the coach have used the median instead of the
mean? Why?
$ in
fundraiser
sales
20
40
40
60
20
400
80
40
Describing quantitative Numerically
why is knowing variability/spread important?
Consider the following test scores:
Student
Test 1
Test 2
Test 3
Test 4
Johnny
65
82
93
100
Will
81
86
88
83
Anna
80
99
72
88
Who is the best student?
How do you know?
Describing quantitative Numerically
spread: iqr
Interquartile Range – the range of values between Quartile
1 (the middle of the first half of data) and Quartile 3 (the
middle of the second half of the data)
 Use this when using MEDIAN as your center
 The box of the box plot – accounts for 50% of the data
 Use this to find OUTLIERS:
If a data value is
< Q1 – 1.5(IQR)
Or > Q3 + 1.5(IQR)
Describing quantitative Numerically
spread:
deviation
-4
-3
+5
-1
0
1
2 3
4
+3
5
6
7
8
x
Sum of deviations = (-4)+(-3)+(-1)+(+5)+(+3) = 0
An average deviation of zero – wouldn’t that
represent that there is no variability?
9
10
11
Describing quantitative Numerically
spread: Mean absolute deviation
 How can we fix the problem?
 Take the absolute value of each distance/deviation and
then find the average
sum of the absolute value
of the deviations from the mean 
 4   3   1  3  5  16
16
average   3.2
5
 So the average distance or deviation from the mean is
about 3 points (above or below).
 This is called the Mean Absolute Deviation, or MAD
Describing quantitative Numerically
spread: standard deviation
 OR another way we can fix the problem:
 Instead of absolute value, take the square of each
distance/deviation and then find the average
sum of squared deviations
 (4) 2   3  12 32  52
 16  9  1  9  25  60
60
average squared deviation 
 12
5
2
Then take the square root to " un - do" the squaring
12  3.4
 This is called the Standard Deviation
Describing quantitative Numerically
spread: standard deviation
Just like mean has two symbols, use μ when the
data is the entire population, and use x when
the data is a sample of the population
Formula for the population’s standard deviation:
( x   )

n
2
Formula for the sample of a population’s
standard deviation:
2
( x  x )
s
n 1
Back to Johnny, Will and Anna . . .
Calculate the standard deviation for each
student.
Student
Test 1
Test 2
Test 3
Test 4
Johnny
65
82
93
100
Will
81
86
88
83
Anna
80
99
72
88
Test Standard
Average Deviation
Student
Test 1
Test 2
Test 3
Test 4
Johnny
65
82
93
100
85
13.2
Will
81
86
88
83
85
2.7
Anna
80
99
72
88
85
10.0
Activity extension:
If on test 5, Johnny, Will, and Anna all earned
a 90. Without calculating, whose standard
deviation would be most affected? Why?
Remember…
 Mean and standard deviation are paired (use when there are
no outliers) …and Median and IQR are paired (use when there
are outliers)
 When interpreting data graphically or numerically, remember
to remind students to write in complete sentences in context
of the problem.
 Keep in mind that students need to know how to compare two
data sets and know what happens to center and spread of the
data when another data point is entered
 All the documents and then some are on the wiki under
Common Core Math 1 - Statistics
 A remediation resource: SAS Curriculum Pathways –
www.sascurriculumpathways.com
Quicklaunch for One Variable Statistics is #5071 and #5072
 Please make sure you post any questions during break.
PROJECT IDEAS
 At your table, discuss what would be a great
summative assessment in the form of a project
for this unit.
 Write down a few ideas on chart paper.
 When finished, put up on the board.
 When you get back from break, do a gallery walk
to see the different project ideas.