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Common Core Math 1 Statistics
WELCOME!!!
Common Core State StandardsReason 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 CC1A (10 days) CC1 (5 days)1 Introduction to Statistics,
Categorical vs. Quantitative DataIntro to Statistics, Categorical vs. Quantitative (clickers); begin Histograms
2 Frequency Tables and Histograms Frequency Tables, Histograms, Dot Plots (do activity with paper clips); use graphing calculator
3 Histograms vs. Dot Plots; Using the Graphing Calculator to make a Histogram
Measures of Center
4 Measures of Center (focus on Median)
Measures of Spread and Outliers
5 Measures of Center (focus on Mean)
Test (have projects due Tuesday)
6 Measures of Spread (focus on Standard Deviation)
7 Boxplots. Outliers; discussion on data
8 Activity: Comparing Data Sets9 Review Using Data10 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
Dotplot
Histogram
Boxplot
Quantitative - Graphically
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)
Let’s collect some data
When Describing quantitative data
from a graphShape
Outliers
Center
Spread
Symmetric/Mound Shaped
Skewed Left(extreme low values)
Skewed Right(extreme high values)
Uniform
Estimate the median The middle value, same number
of values on the left and right side
Estimate the mean The “balancing point” or
average
Is the data spread out? How far is the range? (max – min) Do you think this would have a large or small standard deviation?
Any unusual values, outside the normal trend of data
Histograms 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
Boxplots 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
Quantitative - GraphsWe 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.
Describing NumericallyCategorical Data
Counts Proportions Percents
Quantitative Data 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. medianMeanUse the mean to describe the middle ofset of data that does not have an outlier
μ (with entire population) or (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 dataDisadvantages Affected by extreme values (outliers)
MedianUse the median to describe themiddle of a set of data that doeshave an outlierAdvantages 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.
http://regentsprep.org/REgents/math/ALGEBRA/AD2/measure.htm
n
xx
x valuesdata ofnumber
valuesdata of sum
x
Describing quantitative
Numerically CENTER: mean vs. medianEXAMPLE : 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
sales2040406020
4008040
Describing quantitative Numericallywhy is knowing variability/spread important?
Consider the following test scores:
Who is the best student?How do you know?
Student Test 1 Test 2 Test 3 Test 4
Johnny 65 82 93 100
Will 81 86 88 83
Anna 80 99 72 88
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)
0 1 2 3 4 5 6 7 8 9 10 11
+3
+5
-1
-3
-4
x
Describing quantitative Numerically spread:
deviation
Sum of deviations = (-4)+(-3)+(-1)+(+5)+(+3) = 0An average deviation of zero – wouldn’t that represent that there is no variability?
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
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
2.35
16 average
1653134
mean thefrom deviations theof
valueabsolute theof sum
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
This is called the Standard Deviation
125
60 deviation squared average
602591916
5313)4(
deviations squared of sum22222
4.312
squaring thedo"-un" root to square theThen take
Describing quantitative Numerically spread: standard deviation
Just like mean has two symbols, use μ when the data is the entire population, and use when the data is a sample of the population
Formula for the population’s standard deviation:
Formula for the sample of a population’s standard deviation:
n
x 2)(
1
)( 2
n
xxs
x
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
Activity extension:If on test 5, Johnny, Will, and Anna all earned a 90. Without calculating, whose standard deviation would be most affected? Why?
Student Test 1 Test 2 Test 3 Test 4 TestAverage
Standard Deviation
Johnny 65 82 93 100 85 13.2
Will 81 86 88 83 85 2.7
Anna 80 99 72 88 85 10.0
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.
