Transcript Presenting Data - wsdsecondarymath

The pictures of Statistics
 Central
 Mean
Tendencies -
–
 Median –
 Mode  Statistics -
Statistics are numerical facts and figures.
For instance:
The largest earthquake measured 9.2 on the
Richter scale.
 Men are at least 10 times more likely than women
to commit murder.
 One in every 8 South Africans is HIV positive.
 By the year 2020, there will be 15 people aged 65
and over for every new baby born.

Just like hearing a piece
conversation out of context, some
statistical facts can lead to wrong
conclusions.
What is wrong with this conclusion?
A new advertisement for Ben and Jerry's ice cream
introduced in late May of last year resulted in a 30%
increase in ice cream sales for the following three months.
Thus, the advertisement was effective.
A major flaw in this problem is that
ice cream consumption generally
increases in the months of June, July,
and August regardless of
advertisements. In this case the
increased temperature is probably
more responsible for the increase in
sales than the ad campaign was.
The more churches in a city, the more
crime there is. Thus, more churches
lead to more crime.
Solution
A major flaw is that both increased churches and increased crime
rates can be explained by larger populations. In bigger cities, there
are both more churches and more crime. This problem, In this case
a third variable can cause both situations; however people
erroneously believe that there is a causal relationship between the
two primary variables rather than recognize that a third variable
can cause both.
 The
first step is to learn to think about the
situation.
 Second, look at what is really causing the
correlation.
 Look at the whole picture!
• Central Tendencies
• Draw a plot
To understand the whole
picture of statistics we
start with the raw data
and organize it into forms
to make it more
understandable.
For one variable data we
use dot plots, histograms,
and box plots.
 Bar
Graph –
 Box Plot –
 Dot Plot –
 Histogram  Interquartile Range –
 Lower Quartile –
 Quartile –
 Range –
 Upper Quartile -
Or this:
Also called a
pictograph.
Also, called a
bar graph
Dot plots are representations of the number
of data entries for different categories.
So whether we use dots, pictures or bars
they make the fall into this same category of
a representational statistical plot.
In a histogram:



Area of the rectangles is the most
important part. So group range is
very important for showing the
distribution accurately. (graph
paper is best for this)
The groups must be touching.
No values can be left out and no
overlapping of values in different
groups.
 These
are sometimes called Box
and Whisker plots.
 They are the best for really looking
at the how the data values relate to
the whole picture.
Suppose you were to catch and measure
the length of 13 fish in a lake:
12, 13, 5, 8, 9, 20, 16, 14, 14, 6, 9, 12, 12
A box and whisker plot is based on the medians.
The first step is to rewrite the data in order, from
smallest length to largest:
median
5, 6, 8, 9, 9, 12 ,12, 12, 13, 14, 14, 16, 20
Now find the median of all the numbers.
8.5
Median
Lower Quartile
 The
next step is to find the lower
quartile which is sometimes called the
lower median. This is the middle of the
numbers lower than the median
 In this case the median falls between 8
and 9 so it will be 8.5
14
8.5
Median
Lower Quartile
 Now
Upper Quartile
find the upper quartile or upper
median. This is the middle of the upper
six numbers.
 In this case the upper quartile falls the
two numbers on each side of the upper
median are the same, so the upper
quartile is 14.
 First
you will need to draw a number
line that extends far enough in both
directions to include all the numbers in
your data:
 Draw
vertical lines above the median,
lower quartile and upper quartile.
 Draw
connecting horizontal lines to form
the box.
 Finally, the
whiskers extend out to the
data's lower extreme and upper
extreme, and your done.
But what does it mean? What information about
the data does this graph give you?
We can see from the graph that the lengths of the
fish were as small as 5 cm, and as long as 20 cm.
This gives you the range of the data ... 15.
You also know the median, or middle value was
12cm.
Since the median and quartiles represent the
middle points, they split the data into four equal
parts. In other words:
 one quarter of the fish are less than 8.5 cm.
 one quarter of the fish are between 8.5 and 12 cm.
 one quarter of the fish are between 12 and 14 cm.
 one quarter of the fish are greater than 14 cm.
Statistics Part 3
Outliers and standard deviation
Extremely high or extremely low numbers
compared to the rest of the data are called
outliers. Outliers can represent a problem with
quality control for businesses.
For example, one year I was doing statistics with
peanut M&M’s. I handed a small package out to
each student for them to graph the distribution
of the colors.
One package contained only one M&M.
This aberrant package was an outlier
when compared to the number of candies
in the other packages.
This was clearly a mistake, but it made us
think about some questions.
 What if this happened often?
 Would customers be happy?
 How can outliers like this affect
businesses?
 What can outliers like this mean in other
situations, like test scores?
 Some
times outliers are obvious, but
other times they are not.
 Statisticians use a formula to find them
 First take the interquartile range and
multiply it by 1.5. This will tell you how
long each whisker should be.
 Any data values outside this distance are
outliers.
A cereal company packages bags of granola.
A quality control manager tested 15 random
bags for weight. Are there any outliers in this
data? What does this data mean for your
company?
10.2, 14.1, 14.4. 14.4, 14.4,
14.5, 14.5, 14.6, 14.7, 14.7,
14.7, 14.9, 15.1, 15.9, 16.4
(Hint: start with building a box plot)
To find the Interquartile
Range subtract the lower
quartile from the upper
quartile.
𝐼𝑄𝑅 = 14.9 − 14.4 = 0.5
 Next
find the length of acceptable whiskers,
by multiplying the interquartile range by
one and a half.
0.5 ∙ 1.5 = 0.75
 This
tells how far away from the lower
quartile and the upper quartile we can go
before the data becomes aberrant.
 Anything
outside this range is an outlier.
This means that every bag that has less than 13.65 ounces or
more than 15.65 ounces are not acceptable.
By comparing the data entries we can see which numbers
are outliers.
10.2, 14.1, 14.4. 14.4, 14.4, 14.5, 14.5, 14.6, 14.7, 14.7,
14.7, 14.9, 15.1, 15.9, 16.4
 In
statistics we often look at clusters of
data as well as the extremes.
 It is also very important that we look at
how spread out the data is. Why?
 In a box and whisker plot, we get an idea
of the spread by looking at how stretched
out the box and whiskers are.
 But another way is to look at how close
data is to average is to calculate the
Standard Deviation
 Deviation
means how far from the normal
something is. A small deviation means
you are close to average. A large one
indicates you are really far out.
 The Standard Deviation of a data set is
the measure of how spread out the values
are. A small SD means the numbers are
clustered, but a big SD can mean the
numbers don’t have a close relationship.
 The
formula is easy: it is the square root
of the Variance. So now you ask, "What is
the Variance?"
 The
Variance is defined as: The average
of the squared differences from the
Mean.
Don’t worry it makes more sense after you’ve done it.
You and your friends have just measured
the heights of your dogs (in millimeters):
The heights (at the shoulders) are: 600mm,
470mm, 170mm, 430mm and 300mm.
So now we now the variance is 21,704
Hey, what’s that
funny σ thing?
It’s a lower case sigma.
It’s the symbol for
Standard Deviation.
Standard Deviation:
σ = √21,704 ≈ 147.32... ≈ 147
(to the nearest mm)
So, using the Standard Deviation we have a "standard" way of
knowing what is normal, and what is extra large or extra small.
Rottweilers are larger than average and Dachshunds are
smaller than average.