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Summation Notation
The symbol ∑ means sum. If
, is a set of n numbers (data) then
is the sum of those numbers. We often abbreviate
The expression
represents the average (mean) value of the numbers
If
.
is a sample data set then n is called the sample size.
Example. If a data set is 3, 8, 4, 5, 3, 4, 6 (n=7), compute
a) ∑ x = 3+8+ 4+ 5+ 3+ 4+ 6 = 33
b) ∑ x2 = 32+82+ 42+ 52+ 32+ 42+ 62 = 175
c) ∑ x2 – (∑ x)2 = 175 – 332 = -914
2.3 – Numerical Measures of Central Tendency
Below is a dot plot of some sample data
Where is the center of this distribution?
Common measures of the center (or central tendency) of a set of numbers are
mean, median, and mode
The mean (average) value of a sample data x is called the sample mean and is
denoted by x
x
Sample Mean: x 

n
where n is the sample size (= number of observations)
The “mean (average)” value of a variable x in the whole population is called the population
mean and is denoted by µ. The population mean µ is rarely know or computable. It is a
parameter of a model that mathematically describes the population.
Example. The mean of -1, 0, 2, 5 is
x 
1 0  2  5
 1.5
4
The median of a sample data x the middle value (if n is odd) or the average of
two middle values (if n is even) of observations arranged in ascending order.
Example. a) Find the median of 5, 2, 7, 1, 0
Solution: Sorted data: 0, 1, 2, 5, 7, median = 2
b) Find the median of 5, 2, 7, 1, 0, 4
Solution: Sorted data: 0, 1, 2, 4, 5, 7, median = (2+4)/2 = 3
The mode is the value that occurs most frequently in a data set.
In the case when data consists of large number of different observations we instead
define the modal class, which is an interval (or intervals) in a histogram for these
data that contains the largest number of observations.
Example. a) The mode of the data set 0, 2, 3, 1, 4, 3, 2, 3, 5 is 3
b) For R&D data from Section 2.1 the modal class is [7,8)
Exercise. Given data 1, 2, 7, 10, 2, 4, 4, 13, 2.
a) Find the mean.
b) Find the median.
c) Find the mode.
d) Make the dot plot and mark all the three values.
e) Do a) and b) using TI-83 and Excel
Geometric Interpretation
 Median = value that divides histogram into two equal halves
 Mean = balancing point of a histogram (or a dot plot)
50%
median
50%
mean
2.4 –Measures of Variability
A) 3, 3, 3, 3, 3, 3, 3, 3, 3
NO VARIABILITY
B) 3, 2, 3, 4, 4, 3, 3, 3, 2
SMALL VARIABILITY
C) 2, 6, 4, 1, 3, 0, 2, 5, 4
LARGE VARIABILITY
Common measures of variability are: range, variance, and standard deviation
Range = maximum – minimum
Example. For the above data sets the ranges are:
Range for the set A = 3 – 3 = 0,
Range for the set B = 4 – 2 = 2, Range for the set C = 6 – 0 = 6
Deviation =
x  x = difference between a data value and the sample mean
xx
x
x
Example: For the above data set 0 1 2 3 3 4 4 5 6 the sample mean x  3.1
and the deviation of x = 5 from the mean is x  x  5  3.1  1.9
Sample Variance = s2 = “average” square deviation of data values from their mean
Sample Standard Deviation = s = square root of the variance.
Example. Compute the mean, variance and the standard deviation for the data set C)
2, 6, 4, 1, 3, 0, 2, 5, 4: a) by hand, b) using TI-83, c) using Excel
a) the sample size n = 9.
the sample mean =
the sample variance =
the sample standard deviation =
b) TI-83 commands: STAT - EDIT – [choose] 1:Edit… - ENTER – [enter data in column L1] STAT - CALC – [choose] 1:1-Var Stats… - ENTER – L1 [2nd 1] – ENTER
c) Excel functions:
sample mean =AVERAGE(block),
sample standard deviation = STDEV(block)
Exercise. Given data 1, 2, 7, 10, 2, 4, 4, 13, 2.
a) Find the mean.
b) Find the deviation of x = 2 from the mean
c) Find the range
d) Find the variance s2 and the standard deviation s for this data set (any method)
e) From each data value subtract the mean and divide the result by s, that is for each x
compute
.
Then compute the mean and the standard deviation for the set of z’s.