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Algebra 1 Statistics Part 1 Unit 1.
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Transcript of Algebra 1 Statistics Part 1 Unit 1.
Algebra 1 Statistics Part 1 Unit 1 Todays Objectives Calculate
Mean, Median, Mode from a set of data.
Calculate 5 number summary froma set of data. Calculate Range from
a set of data. Calculate Interquartile Range (IQR)from a set of
data. Common Core State Standard Focus standard for the day: S.ID.1
Represent data with plots ona real number line. (box-plots)
Question: What is the first step in finding the median, andwhat is
the 5 number summary? Measures of Center The MEAN of a data set is
its average.
Use the symbolfor mean. Calculate by adding all the numbers and
divide by how many individualnumbers there are. I have 5 different
numbers that represent the ages of randomly selectedpeople:11, 72,
83, 94, 25 mean: 5 The average age of these 5 people is 57 years
old. The MEDIAN of a data set is its midpoint. That is, half the
data fall above the median and half fall below the median.(50%
above, 50% below) To find the median: Sort data from low to high,
count to middle. The MODE is the most frequently occurring value
Lesson 1.2.1 Rainy Days! For the last 15 years, I have kept track
of the number of rainy days we had in April. The results are below.
Calculate the mean, median, and mode of these data. Here are the
data: 16, 3, 16, 15, 13, 26, 15, 13, 14, 3, 10, 8, 9, 2, 9 Mean:
_________ Median: _________ Mode: __________ Lesson 1.2.1 Rainy
Days! For the last 15 years, I have kept track of the number of
rainy days we had in April. The results are below. Calculate the
mean, median, and mode of these data. Here are the data: 16, 3, 16,
15, 13, 26, 15, 13, 14, 3, 10, 8, 9, 2, 9 Rearranged: 2, 3, 3, 8,
9, 9, 10, 13, 13, 14, 15, 15, 16, 16, 26 Mean: ____11.46_____
Median: ____13_____ Mode: _____3,9,13,15,16_____ Lesson 1.2.1 5
Number Summary Consisting of: Minimum, Q1, Medium, Q3, and
Maximum
Minimum: Smallest value of the sample data Q1: first quartile, this
is the median of the lower half of data [lower 25% of data falls in
this range] Median: technically Q2, middle point of sample data Q3:
third quartile, this is the median of the upper half of data [upper
25% of data falls in this range] Maximum: largest value of the
sample data This 5 number summary can be used to create a boxplot.
(aka boxand whisker plot) Find the 5 number summary, IQR, and
Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:_______
Q1:________Range:_______ Med:_______IQR:________ Q3:________
Max:_______ Lesson 1.2.1 Find the 5 number summary, IQR, and
Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:_______ Med:___13____IQR:________ Q3:_____15___
Max:___26____ Lesson 1.2.1 Measures of Spread: 5 number
summary
RANGE = maximum value minimum value Inter-quartile range (IQR) = Q3
Q1 Talk for next class: Outliers are observations (data points) too
far removed from themain body of data. Any observation above Q x
IQR Any observation below Q1 1.5 x IQR Lesson 1.2.1 Find the 5
number summary, IQR, and Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:_______ Med:___13____IQR:________ Q3:_____15___
Max:___26____ Lesson 1.2.1 Find the 5 number summary, IQR, and
Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:___24___ Med:___13____IQR:____7____ Q3:_____15___
Max:___26____ Lesson 1.2.1 Lets Practice what weve learned!
End of day 1 Lets Practice what weve learned! Todays Objectives
Using the 5 number summary, IQR,and Range to describe the data.
Constructing a box plot. Describing the Center, UnusualPoints,
Spread, Shape (C.U.S.S) Communicate! Communicate! Outliers- what
are they? Calculating Outliers using the 1.5 rule. Common Core
State Standard Focus standard for the day: S.ID.1 Represent data
with plots on areal number line. (box-plots) S.ID.2 Use statistics
appropriate to theshape of the data distribution tocompare center
(median, mean) andspread (interquartile range, standarddeviation)
of two or more data sets. S.ID.3 interpret differences in
shape,center, and spread in the context ofthe data sets, accounting
for possibleeffects of extreme data points (outliers). Question:
What are the 4 main characteristics to describewhen reporting your
findings on a set of data? Find the 5 number summary, IQR, and
Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:___24___ Med:___13____IQR:____7____ Q3:_____15___
Max:___26____ Lesson 1.2.1 Boxplots: Box and Whisker Plot Make a
box plot! Lets make a box plot using the same important
informationfrom the 5 number summary! Answer The mean is 11.5
Data:2, 3, 3, 8, 9, 9, 10, 13, 13, 14, 15, 15, 16, 16, 26 The
median is 13 For the boxplot, we also need Minimum = 2 and Maximum
= 26 First Quartile = 8 and Third Quartile = 15 Lesson 1.2.1
Outliers Outliers are observations (data points) too far
removedfrom the main body of data. Outliers often skew our data. We
can calculate how tofind outliers with the 1.5 Rule for outliers.
Upper Outlier: Any observation above Q x IQR Lower Outlier: Any
observation belowQ1 1.5 x IQR How did this apply to the Rainy
Days?The next slidehas the data again. Lesson 1.2.1 Find the 5
number summary, IQR, and Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:___24___ Med:___13____IQR:____7____
Q3:_____15___Outliers?Upper: Q (IQR) Max:___26____ Lower:Q1
1.5(IQR) Lesson 1.2.1 Find the 5 number summary, IQR, and
Range
Rainy Days! For the last 15 years, I have kept track of the number
of rainy days we had inApril.The results are below. Find the 5
number summary, IQR, and Range Here are the data: 2, 3, 3, 8, 9, 9,
10, 13, 13, 14, 15, 15, 16, 16, 26 Min:___2____
Q1:____8____Range:___24___ Med:___13____IQR:____7____
Q3:_____15___Outliers?Upper: Q (IQR)= 25.5 Max:___26____ Lower:Q1
1.5(IQR)= -2.5 Lesson 1.2.1 Outlier in our data! 26 is an outlier
for our data!
We must take this into account when constructing a boxplot When
constructing a box plot we now only extend the whisker to the
pointof the data that stays within our boundaries for outliers. The
way we represent an outlier on a box plot is just a single point
(dot). Lets make a new boxplot taking this into account! Lets look
at the box plot again! Answer Data:2, 3, 3, 8, 9, 9, 10, 13, 13,
14, 15, 15, 16, 16, 26 The median is 13 For the boxplot, we also
need Minimum = 2 and Maximum = 26 First Quartile = 8 and Third
Quartile = 15 Lesson 1.2.1 C.U.S.S C: Center Median, where is it?
Mean can also describe the center, but is not resistant U: Unusual
data points Outliers! Are there any? We can calculate them S:
Spread Describe the variability of the graph. Range. (largest value
smallest value) S: Shape Is the data clumped in a general location?
Is data stretchingto the right (skewed right). Is the data
stretching to the left(skewed left). LASTLYAlways, ALWAYS describe
data with these 4main points. Communication is key in Statistics!
C.U.S.S put to use with rainy days example:
After we have crunched numbers and calculated all of this
information thatdescribes this quantitative data, we need to
communicate it back into context! My write up: The data recorded
for the number of rainy days in April for the past 15 yearsappears
to have a median number of 13 rainy days with one outlier of 26
rainydays. The data appears to be clumped together with a spread of
24 rainy days;this includes the outlier found in our recorded data.
Center: median number of 13 rainy days Unusual points: with one
outlier of 26 rainy days. Spread: with a spread of 24 rainy days
Shape: the data appears to be clumped together Lets Practice what
weve learned!
End of day 2 Lets Practice what weve learned! Day 3: Activity!
Putting it all together
Todays class will be spent being statisticians and collecting data
from thefield. You will be required to collect data from everyone
in this class! Organize your data and create a presentation of your
data to the class. See hand out! Algebra 1 Statistics Part 2 Unit
S2 Todays Objectives Understand bell curve and theEmpirical rule %
Calculate standard deviation fromVariance Common Core State
Standard Focus standard for the day: S.ID.2 Describe variability
bycalculating deviations from themean Bell Curve Statistics is
about representing data and analyzing it in order to report back
thefindings. Part of representing data is through graphs. Pie
charts, histograms, stem-and-leafplot, and box plots. All of these
graphs can be transformed from one or the other. One of the most
useful visuals in statistics is the Standard Normal Curve. Or
bellcurve Here is what a bell curve looks like, and also a matching
box plot. Notice how the median is centered in the middle of the
bell curve
Notice how the median is centered in the middle of the bell curve.
Couldwe break the box plot up into the percentages of data that
falls betweeneach quartile?? Empirical rule % Similarly to breaking
up our boxplots into quartiles and describing the percentages ofhow
much data falls between each section, we can also break up the bell
curveinto Standard Deviations, we denote standard deviation with
lower case Greek lettersigma Standard deviation describes the
distance away from the center of the bell curve.(mean/median) 1
standard deviation describes that 68% of the data will fall between
+1 and -1standard deviation. 2 standard deviations describes that
95% of the data will fall between +2 and -2standard deviations. 3
standard deviations describes that 99.7% of the data will fall
between +3 and -3standard deviations. Lets take a look at a visual!
Empirical rule standard deviation We can calculate the standard
deviation.
Since we will be sampling from different areas of interest, such as
baseball,medical records, insurance records, cars, machinery,
aircraft, medical devices,household items, agriculture the list
goes on! We need to make sure we aretalking about standard
deviation in context to the problem! Every set of sampledata has
its own unique sample standard deviation. Since we are touching the
basis of statistics at this point we will not worry
aboutdistinguishing between what it means to calculate Sample
Standard Deviationversus the Population Standard Deviation. At this
point we want to get down thebasics, then later down your math
career we will make sure to distinguish betweenthe differences!
Calculating the Standard Deviation
To find the standard deviation all we have to do is take the square
root of thevariancebut to find the variance we have to do a bit of
work! Variance is described as the average squared distance Step 1:
find the mean of your data Step 2: subtract the mean from each and
every data point you have. Step 3: square the differences of each
data point Step 4: add up al the squared-differences, then divide
the sum by 1 this isVariance! Step 5: take the square root to find
the standard deviation Lets look at an example together and walk
through this! A group of 9 elementary school children was asked how
many pets they have. Here are their responses, arranged from lowest
to highest Step 1: Find the mean = =5 So we now know that the mean
number of pets owned by these 9 children is5 pets. A group of 9
elementary school children was asked how many pets they have. Here
are their responses, arranged from lowest to highest Step 2:
Subtract the mean from each and every data point we have. Step 3:
square the differences from each and every point we have. A group
of 9 elementary school children was asked how many pets they have.
Here are their responses, arranged from lowest to highest Step 4:
add up all the squared-differences, then divide the sum by 1 this
is Variance! 2 = 1 = 6.5 Step 5: take the square root to find the
standard deviation =6.5 =2.55 The average amount of pets owned of
this group of 9 Elementary School Children is5 pets with a Standard
Deviation of 2.55 Pets. Rule with pets So back to the standard bell
curveif our mean is the center, and we know thestandard deviation
nowLet me ask you this How many pets could we say 68% of students
sampled have? [ hint: (1) ] How many pets could we say 95% of
students sampled have? [ hint: (2) ] How many pets could we say
99.7% of students sampled have? [ hint: (3) ] REMEMBER STATISTICS
IS ABOUT COMMUNICATION, COMMUNICATION! Exit Ticket: You try! Here
are data from 5 different dogs and their height. Find the standard
deviation by hand. 600, 470, 170, 430, 300 Step1: Mean Step 2:
Difference Step 3: Square Step 4: add and divide by 1 Step 5:
square root Todays Objectives Scatter plots: Positive, no, negative
correlation
Correlation does not implycausation! Two way tables Probability vs.
Conditionalprobability Common Core State Standard Focus standard
for the day: S.ID. 5 Interpret a table that dividesdata into
different categories S.ID.6 Represent data on twoquantitative
variables on a scatterplot S.ID.6 Describe how twoquantitative
variables on a scatterplot are related Scatter Plots A Scatter (XY)
Plot has points that show the relationshipbetween two sets of data.
In this example, each dot shows one persons weight versus their
height. Scatterplots The association between two quantitative
variables can beshown on one graph by plotting data points as
ordered pairs onaxes.Such a graph is called a scatterplot. Scatter
plots do not have connected points If it seems that one variable is
a response to the other, then plotthat variable on the y axis.It is
called the response variable(dependent variable). The x axis then
has the explanatory variable. (independent variable) Temperature C
Ice Cream Sales
14.2 $ 215 16.4 $ 325 11.9 $ 185 15.2 $ 332 18.5 $ 406 22.1 $ 522
19.4 $ 412 25.1 $ 614 23.4 $ 544 18.1 $ 421 22.6 $ 445 17.2 $ 408
The local ice cream shop keepstrack of how much ice creamthey sell
versus the noontemperature on that day. A scatter plot can show
that arelationship exists between twodata sets. Examples Of
Correlations: Pos, none, neg? Correlation does not imply
causation
Think about it. In a study of college freshmen, researchers found
that students who watched TVfor an hour or more on weeknights were
significantly more likely to have highblood pressure, compared to
those students who watched less than an hour of TVon weeknights.
Does this mean that watching more TV raises ones bloodpressure?
Explain your reasoning. Ask yourself. What possible outside factors
could be in play here? Do those factors have more logical reasoning
as to effect blood pressure? Moral of the story Just because there
is a correlation, DOES NOT imply that one variable causes theeffect
of the other! There can be a lurking variable another factor that
could beinfluencing the cause of a variable. You try! Airline
Outsourced Percent Delay Percent Air Tran 66 14 Alaska 92 42
American 46 26 American West 76 39 ATA 18 19 Continental 69 20
Delta 48 Frontier 65 31 Hawaiian 80 70 JetBlue 68 Northwest 43
Southwest United 63 27 US Airways 77 24 The Problem:Airlines have
increasingly outsourced the maintenance of theirplanes to other
companies. Critics say that the maintenance may be lesscarefully
done, so that outsourcing creates a safety hazard. As evidence,
theypoint to government data on percent of major maintenance
outsourced andpercent of flight delays blamed on the airline (often
due to maintenanceproblems). Make a scatterplot that shows how
delays depend on outsourcing. Probability vs conditional
probability
Two way tables Probability vs conditional probability Basics Two
way frequency tables are a visual representation of the
possiblerelationships between two set of categorical data. The
categories are labeledat the top and the left side of the table,
with the frequency info appearing inthe interior cells of the
table. The totals of each row appear at the right, andthe totals of
each column appear at the bottom. If you could have a new vehicle,
would you want a sport utility vehicle or a sports car?
Entries in the body of the table are called joint frequencies.The
cells that contain the sum are called marginalfrequencies. Two Way
Relative Frequency Table
Displays percent or ratios instead of frequency counts. Thesetables
can show relative frequencies for the whole table, forrows, or for
columns. Relative frequencies can be shown as aratio, decimal or
percent. Probability When looking at a relative frequencytable the
percent or ratio is also theprobability of that event happening
over the ENTIRE TOTAL. If asked, What's the probability a
maleselects an SUV? 21/240 If asked, What's the probability afemale
selects an SUV? 135/240 If asked, What's the probability that aSUV
is selected? 156/240 Probability Notice how all the probabilities
have a denominator of 240! Its out of theentire table total! Moral
of the story When asked fora probability that does not have
apreexisting condition look for the specific characteristics
desired in thetable divided by the table total. P Event A = number
of outcomes corresponding to event A Or you can look at it this way
P specific charac = Specific characteristics table total
Conditional probability
When we are calculating the probability of an event occurring given
thatanother event has occurred, we are describing conditional
probability. Certain conditions have been preselected, and now we
much calculate theprobability based on that condition already
happening. When we have conditional probability our denominator
value becomes thecolumn total or the row total depending on which
condition is given. Example: What is the probability of selecting a
sports car given a male? V.S. What's the probability a male selects
an SUV? Conditional probability
Notice how the totals for each box is over the TABLE TOTAL? What if
we knew one of the variables already? What is the probability that
its asports car GIVEN that its a male? Then our probability
changes!! ( = Probability( sports car given that its a male) =
=.65=65% Comparing two different questions
What is the probability of selecting asports car given a male?
What's the probability a maleselects an sports car? (male selecting
an sports car)= =0.1625=16.25% ( = =.65=65% Flashback! On April 15,
1912, the Titanic struck an iceberg and rapidly sank with only710
of her 2,204 passengers and crew surviving. Data on survival
ofpassengers are summarized in the table below Survival Status
Class of Travel Survived Died Total First Class 201 123 Second
Class 118 166 Third Class 181 528 Conditional Probability
1 201/324 3 528/709 1 201/500 P(survived) 500/1317 Survival Status
Class of Travel Survived Died Total First Class 201 123 324 Second
Class 118 166 284 Third Class 181 528 709 500 817 1317 Todays
Objectives Correlation coefficient r
Cover the understanding that r fallsbetween, 1 #4 Linear Regression
(Ax + b)> L1, L2 You will get the linear regression line, and
R-square. Take square root toget r! 2) Compute r value with
website: Lets try The local ice cream shop keeps track of how much
ice cream they sell versus the noon temperature on that day.
Compute the r value with TI 83: Compute the r value with For
Homework: Collect data from your classmates (teacher choice: whole
class / half class /section) Make a scatter plot of your sample
with the graph provided Label axis correctly with a title Calculate
the correlation constant using technology Interpret the correlation
constant and scatter plot; Direction Strength Remember:
communication is important, so report your answer in context ofthe
problem! Todays Objectives Regression Line introduction Line of
best fit
Interpret Slope and y-intercept of a linearmodel in the context of
the data Common Core State Standard Focus standard for the day:
S.ID.6a Practice steps to find the bestline using technology of
your choice S.ID.6b Quantify the goodness of fit of asmall data set
by plotting and analyzingresiduals S.ID. 6c Fit a linear function
for a scatterplot that suggests a linear association S.ID.7
Interpret the slope and the interceptof a linear model in the
context of thedata F.IF.6 Calculate and interpret the averagerate
of change of a function (presentedsymbolically or as a table) over
aspecified interval. Estimate the rate ofchange from a graph. Draw
a quick sketch of three scatterplots:
Warm up/ pop quiz Draw a quick sketch of three scatterplots: Draw a
plot with r .9 Draw a plot with r -.5 Draw a plot with r 0
Temperature C Ice Cream Sales
14.2 $ 215 16.4 $ 325 11.9 $ 185 15.2 $ 332 18.5 $ 406 22.1 $ 522
19.4 $ 412 25.1 $ 614 23.4 $ 544 18.1 $ 421 22.6 $ 445 17.2 $ 408
The local ice cream shop keepstrack of how much ice creamthey sell
versus the noontemperature on that day. Algebra Line of Best
Fit
We can draw a Line of Best Fit on our scatter plot: When creating a
line of best fit we try to have the line as close as possible to
all points, and as many points above the line as below. In Algebra
the Line of best fit comes in the form=+. Statistics Regression
Line
In Algebra our line is known as a line of best fit In statistics,
this is called a regression line! A line that describes how a
response variables y changes as an explanatoryvariable x changes.
We often use a regression line to predict the value of y for agiven
value of x. Formulas for Regression Line
The Regression line is linear, so it follows the form y = mx + b In
Statistics, we say =+(Pronounced y-hat) In this context,is called
the predicted value WARNING: We are entering predicting statistics,
using the correctnotationis very important!! Variable breakdown in
Algebra terms: ... is the predicted value a is the y-intercept b is
the slope =+ The Meaning of Slope In a simple algebraic equation
such as, = 2 + 17, what isthe real meaning of the slope? For every
increase in x of 1 unit, y increases by 2 In the function = 2 + 17
what is the meaning of the yintercept? It is the value y takes on
when x = 0 In statistics ifthe regression line is =17+2 What is the
slope? What is the y-intercept? Example Some data were collected on
the weight of a male white laboratory rat for the first 25weeks
after its birth. A scatterplot of the weight (in grams) and time
since birth (inweeks) shows a fairly strong, positive relationship.
The linear regression equation modelsthe data fairly well: = () 1)
What is the slope of the regression line? Explain what it means
incontext 2) Whats the y intercept? Explain in context 3) Predict
the rats weight at 16 weeks Todays Objectives Interpret 2
value!
Fit a linear regression equation withthe appropriate graph Common
Core State Standard Focus standard for the day: S.ID.6a Practice
steps to find thebest line using technology of yourchoice S.ID.6b
Quantify the goodness of fitof a small data set by plotting
andanalyzing residuals S.ID. 6c Fit a linear function for ascatter
plot that suggests a linearassociation S.ID.7 Interpret the slope
and theintercept of a linear model in thecontext of the data
remember our Ice cream sales depending on temperature
example?
Would you say our regression line drawn generally represents each
data point, giveor take some? Regression lines and scatter
plots
When looking at a scatter plot someone could draw any line.Perhaps
a lineshifted up, shifted down, more left, more right, but which
line is best? We can create a regression line to represent our
data, and we can alsomeasure how accurately that line represents
the data: our numerical valuethat indicates the strength of our
line is 2 much like probability, our 2value falls between0