STAT-151 Lecture Notes - Lecture 3: Quartile, Negative Number, Box Plot
22 Jun 2018
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1. Index and Summation Notation
Numerical Descriptive Measures
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quantitative numerical summaries
descriptive measures.
subscript
index notation
entire
112 145 149 175 181 183 188 189 205
= 112 = 145 = 149 = 175
= 181 = 183 = 188 = 189 = 205
= ++++++++
= 112 + 145 + 149 + 175 + 181 + 183 + 188 + 189 + 205 = 1527
Graphical representation of data (categorical or quantitative) is a nice was
to quickly see trends in the data.
Another approach for data is to use or
Some descriptive measures are formulas with summing the data.
The summation notation is a way of abbreviating the sum of some quantity.
Suppose we have a set of data
and we need to sum this data.
Use the letter with a to represent the data values.
This is called .
For the above data, we could use
To talk about an unspecied member of the data set, we could refer to
(read sub or subscript ) where could be any of the values from
1to9.
Then we would think of the letter (without any subscript) as referring
to the data set.
Then the notation would be the sum of all those values.
This expression is read as summation of .
With the above data
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= 112 +145 +149 +175 +181 +183 +188 +189 +205 = 265735
= (112 + 145 + 149 + 175 + 181 + 183 + 188 + 189 + 205) = 1527 = 2331729
=( ) ( )
21 25 02030002
= 2+1+( 2)+5=6
= 02+03+00+02=07
= 2 +1 +( 2) +5 =34
= 02+03+00+02=017
= 2 02+1 03+( 2) 00+5 02=04+03+0+1=17
=6 07=42
=( )( )
The symbol is the upper case Greek letter sigma (analogous to in the
Latin alphabet).
It is called the
is a variable, but a symbol that means sum whatever follows over
all possible values of the index.
can be followed by more complicated expressions - the expressions are
always evaluated before the summing step.
For example, using the above data
Notice the squaring is done rst. This expression is the sum of squares.
If we wanted the square of the sum, we would use parenthesis:
Notice that - the order matters! (in fact,
for data set and usually the sign applies).
can also be used with more than one variable.
If represents the set and represents the set
then
Notice that in , multiply rst, then sum (i.e. this is the sum of
products).
If we wanted the product of sums, we would have to write
Notice that - the order matters!
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1.
2.
3.
3.1. Mean
2. Doing Calculations
3. Measures of Centre
Since numerical summaries involve calculations that often need to be rounded
off, we need some rules about rounding.
Do not perform rounding until the computation is complete; otherwise,
substantial round-off error can result. This is particularly important with
fractions that have small numbers in their denominators!
Number of digits to round off at: round nal answers that contain units (e.g.
metres, seconds pounds) to one more decimal place than the raw data.
NOTE: we may not always strictly adhere to this rule, so consider it more
of a guideline.
Once a desire precision has been chosen round off (e.g. third digit after the
decimal point), we use the following rule to determine the last digit: If
the next digit is 0 to 4, leave the last desired digit as calculated (e.g.,
45.23459894 is rounded to 45.23 if we are rounding to two digits after
the decimal point). If the next digit is 5 to 9, add one to the last digit
calculated and perform carries if necessary (e.g. 45.23459894 is round
to 45.235 if we are rounding to three decimal places, 45.23459894 is round
to 45.23460 if we are rounding to ve decimal places,). NOTE: there are a
couple of calculations where we do use this rule - these will be indicated
when they are encountered.
We often represent a data set by numerical summary measures, usually
called the typical values.
A measure of central tendency gives the center of a histogram or a fre-
quency distribution curve.
The , also called the arithmetic mean, is the most frequently used
measure of centre.
The mean for ungrouped data is dened as follows:
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Document Summary
Graphical representation of data (categorical or quantitative) is a nice was to quickly see trends in the data. Another approach for descriptive measures. quantitative data is to use numerical summaries or: index and summation notation. Some descriptive measures are formulas with summing the data. The summation notation is a way of abbreviating the sum of some quantity. Suppose we have a set of data and we need to sum this data. Use the letter with a x subscript to represent the data values. For the above data, we could use x. To talk about an unspeci ed member of the data set, we could refer to (read sub or subscript ) where. 1 to 9. x i could be any of the values from x x i i i. Then we would think of the letter to the data set. entire x (without any subscript) as referring. Then the notation x i would be the sum of all those values.
