Download Grouped Data

Class Schedule
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Class next week
Holiday the next week (Monday schedule)
Exam the next week
So only one more class before the first
exam
• Will cover Chapters 6, 3, 2. Only material
included in class as shown in Course
Outline.
Class 3
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Return homewk from first class
Get soc sec # from 2 students
Review Assignment 2
Collect Assignment 2
Quiz
• Continue Ungrouped Data
• Start Grouped Data
Very Important Hint for Excel
• I will be putting models for calculation on
the web site.
• You will see the values, not the formulas
• BUT, you can change to view the formulas
• Hold down control and hit the accent key.
• It’s the same key, upper left of the
keyboard, that we use for the tilde ~.
Copying Formulas
• To copy to the next cells, highlight the cell
and drag the little box in the lower right of
the cell.
• Remember that the cells to be used in the
formula will be adjusted
• To copy the value in the cell not the
location, use a $ before the column letter
or before the row number or before each.
Formatting Excel
• To make a column wide enough for the
label or values, double-click on the right
border
• To highlight cells adjacent to each other,
click on one, hold down shift key and
move to the other cells.
• To underline a cell, not just the letters in it,
use the little box on the toolbar
• To merge several cells and center the
label, use the little box with 2 arrows
Excel Doing All the Work
• You now know the summation sign on the
toolbar. Click on the little arrow next to it.
We will find lots of “functions” there.
• Find the Mean. Click on an empty cell,
probably the one just under the mean you
already calculated. Click on the functions
arrow.
• Click Average & highlight the column of
ages, Enter.
• You should have the same answer.
To Calculate the Variance
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Put ages in a column & get the sum
Make a labeled cell for n
Make a labeled cell for n-1
Put label “Mean” & under it put
=, click on the cell with sum, then /, click
on cell with value for n, Enter.
Deviations from the Mean
• Go to the column next to the values, label
it X-X
• In the top cell, put = ,click on the cell with
the first value, put minus sign, put in the
row and column for the mean. This last
value needs dollar signs, e.g. $b$4.
Hit enter.
• Highlight that cell and pull down the corner
box.
• When all cells are filled, go to below the
column and get the autosum.
Take the squares of the dev’ns
• Label the next column “squares”
• In the top cell, put =, click on the top
deviation, put *, click on the top dev’n
again.
• Pull down to all cells in the column
• Take an autosum.
Variance
• The sum of the deviations squared,
divided by
n-1is the variance
• So in an appropriate cell, put =, highlight
the cell with the sum, /, highlight the n-1
cell
• Put the label “variance” or s2 next to it.
Variance
• Click cell under the variance you
calculated
• Click Autofunction (the little arrow by the
summation sign)
• Click More Functions & In select a
category, choose statistical
• Go down to VAR
• Next to the Number 1 box, click on the
little graph, highlight the ages data, Enter,
Enter.
• Should have same answer.
Standard Deviation
• May be the most widely used measure of
dispersion
• Related to the Variance
• It’s just the square root of the variance.
Root-Mean-Square
• Descriptive name for the St Dev
• (∑(Xn-X)2/n-1)-1/2
• We used the square to get rid of neg signs in the
sum but then we go back to the same range by
taking a square root later.
• Go back to excel and get the standard deviation
by taking square root and by letting excel do it
directly from data
Standard Deviation
• Find a suitable empty cell.
• Go to autofunctions and choose math &
trig functions and click on SQRT.
• Highlight the variance so that it will be
used by the function.
• Put the label st. dev., or s, in an empty
cell above or next to your answer
Autofunction Standard Deviation
• Choose another appropriate empty cell
• Label it
• In the empty cell, go to autofunction,
choose statistical, click on STDEV.
• Choose the range of the original data.
• Should get same answer as the calculated
one.
To Compare s from 2 Samples
• Use Coefficient of Variation
• s/X times 100. Answer is a percentage.
• Why use CV?
CV
• Removes differences due to units of
measurement
• We might want to know whether serum
cholesterol levels, measured in mg/100ml,
are more variable than body weight,
measured in pounds
Another Advantage of CV
• Variance or st.dev. if applied to 2 groups
with very different means may give
misleading idea of variability
• Wts of 11year-old boys compared to
weights of 25 yr-olds
11’s, X = 80, s = 10
25’s, X = 145, s = 10
Are they equally variable?
• CV’s = 12.5% & 6.9%
• Young boys wt more variable
Finding the Quartiles
• Have to find locations for Q3 and Q1
• For Q3 loc, find n*3/4
• If it’s not an integer, take the next higher
whole number.
• Find it in the array. That is Q3
• For Q1 loc, find n/4. If it’s not an integer,
take the next higher whole number.
• Find it in the array. That is Q1
Interquartile Range
• Q3 – Q1
• Use autofunction to check your answer.
Percentiles
• Location = n * p/100
• e.g., the 90th percentile is located at
n * 90/100. If its not an integer, go to the
next higher integer.
• Pick that value from the array
• 10% of the values will be higher than P90
and 90% of the values will be lower.
Special Problem
• Biostatistics books give these directions
for finding the percentiles
• However, excel autofunction for percentile
uses an interpolation method
• Our answers will not agree
End of Ungrouped Descriptives
• Continue these topics using Grouped Data
Grouped Data
Grouped Data
We do this all the time
Change in my pocket
3 dimes, 2 nickels, 4 quarters, 6 pennies
(3X10) + 2(5) + (4X25) + (6X1)
Sum = $1.46
Group Mean or Weighted Mean
Average Age of Dr. Jones’ Patients
12, 14, 15, 14, 8, 7, 8, 9, 14, 12
n = 10
∑ = (2X12) + (3X14) + (2X8) + 7 + 9 + 15
X = (24 + 42 + 16 + 31)/10
= 113/10
= 11.3
Frequency
• No. of times each value occurs is called its
frequency
• Instead of n, use ∑fi
Grouped Variance
• Use exactly the same concept as for
Grouped Mean
Types of Data
• Before we continue with grouping, let’s go
back and look at types of data
• Ch. 2
Types of Values
• Nominal
• Ordinal
• Ranked
• Discrete
• Continuous
Nominal
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Comes from “name”
Qualitative
Classes: diabetes, asthma …
Types or categories, may be coded in
numbers: diabetes =1, asthma = 2 …
• May be completely qualitative or semiquantitative, e.g over 45, over 65 …
• In statistics, use the frequency of
occurrence of each class
Ordinal
• When classes progress whether in
ascending order or descending order
• Staging of Cancer
Stage 1 is least serious. Each more
serious. Stage 2, 3 to Stage 4
• An order exists but there is no intrinsic
magnitude, e.g. the difference between
Stage 2 and Stage 3 may be much
greater than the difference between
Stage 3 and Stage 4
Ranked
• Very similar to ordinal data
• Difference is that the order depends on a
quantitative difference
• e.g. Rank students according to highest
average
• Rank diseases according to number of
deaths caused by each
• But still, differences between the
sequential ranks may not be equal
Quantitative Data
• Has intrinsic magnitude
• A count or a weight
• Types: Discrete & Continuous
• Types: Interval & Ratio
Discrete vs Continuous
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Discrete
Eggs. We quantify them by counting.
Number of births. Cannot be a fraction.
Continuous
Butter. We quantify it by weight.
Can be fractional
We can make differences infinitesimally
small
Continuous
Can make intervals infinitesimally small
Height: 5 feet 6 inches, or 5 ft. 6 1/4 in.
or 66.21 inches
Ages: Even if given in years, it is a
continuous variable. 10 yrs or 10 ½ yrs
or 122 months or even months, days, etc.
Interval Data
• Differences are equal
• Temperature
Difference between 20 and 30 degrees
equals difference between 30 and 40
• These do not have a real zero
Interval vs Ratio Data
• This distinction not made in your text.
Interval
• Temperature. The scale we use does not
have a true zero. Intervals are equal but
ratios are not. 400 is not twice as warm as
200.
Ratio
• Weight or height. They do have a true
zero. 4 inches is twice as long as 2
inches.
Data Presentation
• Start with Frequency Tables.
• Next class, graphs.
Frequency Distribution
• Group values within intervals
• All intervals of equal size
• Intervals contiguous, not overlapping
• Not too many groups, not too few
Look at a Frequency Distribution
• Pagano page 12, Table 2.6
• How many intervals?
• Does it look like a good number?
• Look at the distribution of data. Is it clear?
Interval Sizes
• Table 2.6 of Pagano
• Look at left hand column. LL of first
interval is
• 80
• LL of second interval is
• 120
• What is the interval size
• 120-80
Class Limits
• We can call the groups “classes” as well
as “intervals”.
• Why doesn’t the first class go from 80 to
120?
• Would be ambiguous. Where would we
put levels of 120, in the first class or the
second?
Using a Frequency Distribution
• Can we use it to calculate measures of
central tendency?
• Measures of dispersion?
• Yes to both questions, of course.
• But how, we no longer have the actual
values.
• Usually, use midpoints of each class
Midpoint of Intervals
• Use midpoint as though it were the actual
value for the individuals in the interval
• Calculate midpoint: (LL + UL)/2
• (80 + 119)/2 = 99.5
• For each of the next classes, just add 40
to 99.5.
Write out Mid-pts
• Or even re-write the table using mid-pts
• 99.5, 139.5, 179.5, 219.5, 259.5, 299.5,
339.5, 379.5
• Use this to find the mean
Mean from F.D.
• Mid-pt., mi, times frequency, fi, within that
class
• Take the sum
• Also take the sum of the frequencies.
• Mean = ∑ fi, mi, / ∑ fi,
• Before we do an example, let’s take a look
at calculating variance or st. dev.
Variance from Grouped Data
• From ungrouped data ∑(Xn-X)2/n-1.
Review this. Does it make sense?
• For grouped data (∑fi(mi-X)2/(∑fi-1)
• What did we do?
• Substitute m for X and the sum of fi for n.
How get Standard Deviation?
• Just take square root of the variance.
Other Measures
• Mode very easy, just the class with highest
frequency.
• Range, just highest limit minus lowest limit
• Median and quartiles, I think they’re not in
your book for grouped data. If we don’t
come across it, will omit.
Calculating from Frequency
Distributions
• Go to Course Outline on Internet.
• Click on Calculations from Grouped.
End of Class 3
• Hope we get this far.
• Check assignments if we do not get this
far in Class 3.
• You only have to do the topics covered in
class.
Get a Data Set
• Later we will use the cd from back of book
• For now, go to our course outline, lect 3,
data set 1, death rates. Click & open.
• Save onto your computer hard drive. Put
it into a place you can find again.
• At home, do this also.
• In fact, save under two names: one call
death_crude”, other call “death_freqdist”
Making the Frequency Distribution
• Make another copy of the data. Highlight
both columns & pull the copy square.
• Insert a column between the raw data and
the new data.
• Make an array. Highlight the column of
data. Click on the Sort Ascending & when
it asks about the adjoining data, click to
include it.
Making a Frequency Table
• Have to get data. Take cd from back of
book.
• Put in cd slot. If it doesn’t open
automatically, go to My Computer &
choose that drive.