Yesterday’s Policies & Priorities: Are They Right for Tomorrow?
Yesterday’s example these are prices for Internet service packages find the mean, median and mode...
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Transcript of Yesterday’s example these are prices for Internet service packages find the mean, median and mode...
Yesterday’s example
these are prices for Internet service packages find the mean, median and mode determine what type of data this is create a suitable frequency table, stem and leaf plot
and graph13.60 15.60 17.20 16.00 17.50 18.60 18.7012.20 18.60 15.70 15.30 13.00 16.40 14.3018.10 18.60 17.60 18.40 19.30 15.60 17.1018.30 15.20 15.70 17.20 18.10 18.40 12.0016.40 15.60
Answers to yesterday’s problem Mean = 494.30/30 = 16.50 Median = average of 15th and 16th numbers Median = (16.40 + 17.10)/2 = 16.75 Mode = 15.60 and 18.60 bimodal What type of data? numerical, so at least
Interval data. It has an absolute starting point, so it is ratio data
Given this, a histogram is appropriate
Frequency Table
Class Interval Frequency
12.00 – 12.99 2
13.00 – 13.99 2
14.00 – 14.99 1
15.00 – 15.99 7
16.00 – 16.99 3
17.00 – 17.99 5
18.00 – 18.99 9
19.00 – 19.99 1
Stem and Leaf Plot
Stem Leaf
12. 20 00
13. 60 00
14. 30
15. 60 70 30 60 20 70 60
16. 00 40 40
17. 20 50 60 10 20
18. 60 70 60 10 60 40 30 10 40
19. 30
Histogram
How many class intervals?
What does the height of each bar mean?
What does the histogram tell us about the data?
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10 12 14 16 18 20 22 24
Price
Internet Prices Histogram
Variables
In mathematics, a variable is a symbol denoting a quantity or symbolic representation. In mathematics, a variable often represents an unknown quantity.
In statistics, variables refer to measurable attributes, as these typically vary over time or between individuals. Variables can be discrete (taking values from a finite or countable set), continuous (having a continuous distribution function) or neither. Temperature is a continuous variable, while the number of legs of an animal is a discrete variable.
Variables are often contrasted with constants, which are known and unchanging. (Wikipedia, 2004, 2008)
The Two Types of Variables Independent Variable
a variable whose values are arbitrarily chosen placed on the horizontal axis time is always independent (why?)
Dependent Variable a variable whose values depend on the
independent variable placed on the vertical axis
Scatter Plots
a graphical method of showing the joint distribution of two variables where each axis represents a variable and each point on the graph indicates a pair of values
may show a trend a trend indicates a correlation that may be strong or
weak, positive or negative, linear or non-linear
What is a trend?
a pattern of average behavior that occurs over time
a general “direction” that something tends toward
E.g., there has been a trend towards increasing costs in Canada
need two variables to exhibit a trend
An Example of a trend
U.S. population from 1780 to 1960
what is the trend?
is the trend linear?
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PearlReedandKish1940_USpopulationfrom17901940_year1780 1800 1820 1840 1860 1880 1900 1920 1940 1960
019 Scatter Plot
Line of Best Fit
the line of best fit is a line which best represents the trend in the data and is used for making predictions
these can be drawn by hand but there are also methods for mathematically calculating them (median-median and least squares methods are examples that we will study)
gives no indication of the strength of the trend (use the r or r2 value)
An example of the line of best fit this is temperature
data from New York over time, with a median-median line added
what type of trend are we looking at?
see p35 for method for creating a median-median line
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StateofNewYorkHistoricalTemperatureData_winters...1900 1920 1940 1960 1980 2000
Attr2_meantemp = 0.0230StateofNewYorkHistoricalTemperatureData_winterseasonmeanof40wea_ - 21.4
048 Scatter Plot
Creating a Median-Median Line Divide the points into 3 symmetric groups
If there is 1 extra point, include it in the middle group If there are 2 extra points, group one in each end
Calculate the median x- and y-coordinates for each group and plot the median point (x, y)
If the median points are on a straight line, connect them
Otherwise, line up the two outer points, move 1/3 of the way to the middle point and draw a line of best fit
Trends in Data Using Technology
Chapter 1.4 – Trends in Technology
Mathematics of Data Management (Nelson)
MDM 4U
Categories of Correlation correlation scatter plots can
be positive or negative, strong or weak
try looking at the examples on this website to help you understand (see Correlation Picture and Regression Line):
http://www.seeingstatistics.com/seeing1999/gallery/CorrelationPicture.htmlhttp://www.seeingstatistics.com/seeing1999/gallery/CorrelationPicture.html
Regression a process of fitting a line or curve to a set of
data if a line is used, it is linear regression if a curve is used, it may be quadratic
regression, cubic regression, etc. why do we do this? what can we do with the resulting function? http://www.seeingstatistics.com/seeing1999/
gallery/CorrelationPicture.html
Correlation Coefficient
the correlation coefficient r is an indicator of the strength and direction of a linear relationship r = 0 no relationship r = 1 perfect positive correlation r = -1 perfect negative correlation
r2 is the coefficient of determination if r2 = 0.42, that means that 42% of the variation in
y is due to x
Residuals a residual is the vertical
distance between a point and the line of best fit
if the model you are considering is a good fit, the residuals should be small and have no noticeable pattern
why?
y
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x1 2 3 4 5 6 7 8 9
y = 0.0804x + 3.5; r^2 = 0.021
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Collection 1 Scatter Plot
http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
Creating a Median-Median Line Using Technology Click on the wiki Right-click the file armspan_v_height_4_ med-
med.ftm and save to your M:\ or USB drive Open the file Create a scatter plot for each set of data Right-click and select
Median-Median Line Least Squares Line (to see r2 value) Make Residual Plot
References
Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page
The Power of Data
Chapter 1.5 – The Media
Mathematics of Data Management (Nelson)
MDM 4U
There are 3 kinds of lies: lies, damn lies and statistics.
‘4 out of 5 dentists surveyed would recommend sugarless gum to their patients who chew gum. ’
In small groups discuss how this statistical statement could be misleading
Trident conclusions
How many dentists did they ask? 5? 4 out of 5 is convincing but reasonable
5 out of 5 is preposterous 3 out of 5 is good but not great
Recommend Trident over what? Chewing sugared gum? Doing nothing?
Is Trident the “best” sugarless gum? What competitors and variables were considered?
What did the 5th dentist recommend?
“More people stay with Bell Mobility than any other provider.” In small groups, discuss:
1) What variables would be recorded in this study?
2) How could the data be used to arrive at this conclusion falsely?
1) What variables would be recorded in this study?
Number of Bell Mobility subscribers Number of renewed contracts Definition of renewed contract? Renewal during OR upon completion of contract? Contract Length Contract Type (business, home, bundle)
2) How could the data be used to arrive at this conclusion falsely? Does not specify how many more customers stay
with Bell. e.g. Percentage of customers renewing their plan:
Bell: 30% Rogers: 29% Telus: 25% Fido: 28% Did they compare percentages or totals? What does it mean to “stay with Bell”? Honour entire
contract? Renew contract at the end of a term? Are early terminations factored in? If so, does Bell
have a higher cost for early terminations? Competitors’ renewal rates may have decreased
due to family plans Does the data include Private / Corporate plans?
How does the media use (misuse) data? To inform the public about world events in an
objective manner It sometimes gives misleading or false impressions
to sway the public or to increase ratings
It is important to: Study statistics to understand how information is
represented or misrepresented Correctly interpret tables/charts presented by the media