Transcript five number summary

STATISTICS
SOME BASIC STATISTICS
• MEAN (AVERAGE) – Add all of the data together and
divide by the number of elements within that set of data.
• MEDIAN – The middle number in a set of data that has
been arranged from least to greatest.
• MODE – The number that appears the most often in a
set of data. If no element appears more than once the
set is set to have NO MODE. If there is a tie then you list
all of the numbers that are tied as the mode.
M & M & M EXAMPLE
• Lets see if we can calculate the mean,
median, and mode for:
2, 3, 5, 6, 7, 7, 12
• First the mean. We need to find the sum
of our list and then divide by the number of
numbers in our list.
2 + 3 + 5 + 6 + 7 + 7 + 12 = 42
42 / 7 = 6 (Mean)
M & M & M EXAMPLE
2, 3, 5, 6, 7, 7, 12
• Now for the median. The median is the middle
of our list. Sometimes it is easy to tell by looking
but if not I like to use the following calculation.
• Take the number of numbers in our list and add
1 to it. Then divide by 2. That tells you that the
number in that position in the list is your median.
• For this example there are 7 numbers so I take 7
+ 1 which is 8. I then divide by 2 which gives me
4. So the 4th number in our list is the median.
Here it is 6!
M & M & M EXAMPLE
2, 3, 5, 6, 7, 7, 12
• Finally the mode. The mode is the
number that appears most often in the list.
In our example it is 7. Remember if there
is a tie you have to list all of them and if no
number appears more than once we say
there is No Mode.
FIVE NUMBER SUMMARY
WHAT IS IT?
• A five number summary is a statistical tool
used to quickly summarize and gain
insight about a set of data.
• The five number summary consists of the
upper and lower extremes, the median,
and the upper and lower quartiles.
SO WHAT DOES THAT MEAN?
• The upper and lower extremes are the biggest
and smallest numbers that occur in the set.
• The median is the middle term when the data is
arranged from least to greatest.
• The upper and lower quartiles are the median
(middle) of the upper and lower halves of the
data, respectively.
HOW DO I FIND IT?
• Lets try an example. Consider the set
(1,3,4,5,6,7,9). Make sure your list is in
numerical order from smallest to biggest!
LOWER AND UPPER EXTREMES
• The lower and upper extremes are simply
the smallest and biggest numbers from
your set. In our example the smallest
number is 1 and the biggest number is 9.
• 1, 3, 4, 5, 6, 7, 9
MEDIAN (THE MIDDLE)
• The way I use to find the middle of a set of numbers is to
count how many numbers are in your set. In our
example there are 7 numbers.
1, 3, 4, 5, 6, 7, 9
• Next I add one to that number and divide by 2.
So 7 + 1 = 8 and 8 / 2 = 4.
• That means that the 4th number in our list is the middle.
That would be 5!
• 1, 3, 4,
5, 6, 7, 9
WHERE ARE WE?
• So far we have found the smallest and
biggest numbers in our list along with the
middle.
• 1,
3, 4,
5
, 6, 7, 9
UPPER AND LOWER QUARTILES
• 1,
3, 4,
5
, 6, 7, 9
• You will notice that the median (the middle) cuts
our list into two parts. The only remaining thing
to do is find the upper and lower quartiles. The
lower quartile is simply the middle of the lower
half and the upper quartile is the middle of the
upper half.
UPPER AND LOWER EXTREMES
(CONTINUED)
• 1,
3, 4,
5
, 6, 7,
9
• 1, 3, 4 represents the lower half of our data.
• The middle is pretty easy to determine just by looking but
if need be we can use the formula we used before.
Count the numbers in our list (3), add 1 and divide by 2.
• 3 + 1 = 4 and 4 / 2 = 2. That means the second number
in the list (3) is our lower quartile.
• 6, 7,
9 represents the upper half and through the same
process we find the upper quartile is 7.
SUMMARY
• So for our example the five number summary is:
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Smallest Number: 1
Lowest Quartile (Middle of the lower half): 3
Median (Middle): 5
Upper Quartile (Middle of the upper half): 7
Biggest Number: 9
• A little side note for you. Many of you have worked with the
box and whisker plot before. The box and whisker is a
graphical display of the 5 number summary!
ONE MORE???
• Find the five number summary for the set
(1,3,4,6,8,9,10,14).
– What is different about this example?????
• The smallest and biggest numbers are
easy enough but in this example the
middle falls in between two numbers.
– 1, 3, 4,
6, 8, 9, 10, 14
– In this case we would need to take the
average (mean) of the two numbers that it
falls between. In this example the middle
would be the average of 6 and 8 which would
be 7.
1, 3, 4, 6, 8, 9, 10, 14
•
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Now for the middle of the lower and upper halves! You can see that the
lower half and the upper half both consist of 4 numbers so the middle of
each half falls in between numbers again.
1, 3, 4, 6
The middle of the lower half falls in between 3 and 4 so we would need to
take the average between 3 and 4 which gives us a lower quartile of 3.5.
8, 9, 10, 14
The middle of the upper half falls between 9 and 10. Thus the upper
quartile would be the average of 9 and 10 which would result in the upper
quartile being 9.5.
So in summary for this example
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Smallest Number: 1
Lower Quartile (Middle of Lower Half): 3.5
Median (Middle): 7
Upper Quartile (Middle of Upper Half): 9.5
Biggest Number: 14