Download 12-1 Define and Use Sequences and Series

11-1 Find Measures of Central Tendency and Dispersion
Objective: To describe data using statistical measures.
Probability and Statistics Standards 6.0 and 7.0
Name:______________________
* VOCABULARY
1. Statistics: Collecting, organizing and analyzing data and making meaningful decisions based
on your finding
2. Measure of central tendency: A number used to represent the center or middle of a set of data
values. Ex) mean, median, and mode
3. Measure of dispersion: A statistic value that tells you how dispersed (or spread out) data
values are. Ex) range, variance and standard deviation
4. Standard deviation () : A measure that describes the typical difference (or deviation) between
a data value and the mean
5. Outlier (= Extreme value): A value that is much greater than or much less than most of the
other values in a data
A local real estate office publishes monthly newsletters. This month’s newsletter states that the
average sale price of homes in Castro Valley last month was $565,000. However, there is more than
one meaning of the word “average” in this context. The office may be giving the mean sale price, or
might actually be giving the median price. Which is the most appropriate?
* Measures of center or central tendency include the mean, median and mode, each of which is used
to describe the typical value of a set of data. They are determined differently and used in different
circumstances. The table below lists these measures:
Use
Mean
Median
Mode
When
The data are spread out and you want an average of values
The data contain outliers
The data are tightly clustered around one or two values
The mean is found by dividing the sum of the data by the number of values in the data.
The mean is denoted by ______, which is read as "x-bar."
The median is the middle value of the data when all values are put in ascending or descending order.
(If there is an even number of values, find the mean of the two middle values.)
The mode is the data value or values that occur most often.
There may be ___ mode, _____ mode, or ________________ modes.
Ex. 1: Find the mean, median and mode of the following set of data:
{ 42, 35, 39, 40, 38, 45, 35 }
Mean:
Median:
Mode:
You Try: Find the mean, median, and mode of the data set. (*Line up the numbers first!)
17, 14, 12, 12, 13, 16, 12, 14, 14, 10
Algebra 2 Section 11-1 page 1
Another type of statistical measure is measures of variation, also called measures of dispersion or
measures of spread. They measure how scattered a set of data is. Common measurements of spread
include range, variance and standard deviation.
The range of a data set is the difference between the greatest and the least data values.
The standard deviation describes the typical difference (or deviation) between the data values and the
mean.
The variance is the square of the standard deviation.
To find the standard deviation or  (sigma) of a set of n values, x1, x2, . . . , xn,
( x1  x ) 2  ( x 2  x ) 2  . ..  ( x n  x ) 2
whose mean is x , use the following formula:  
n
*How to find Standard Deviation:
1. Calculate the mean ( x ).
2. For each number, subtract the mean. Square the result.
3. Calculate the mean of those squared differences. This is the variance.
4. Take the square root of that to obtain the standard deviation.
(standard deviation is always positive value!).
Ex. 2: Find the variance and standard deviation of the data set {10, 9, 6, 8, 7 }
Step 1: Find the mean.
Step 2: Find the variance. (use the standard deviation formula)
Step 3: Find the standard deviation. (How is the variance related to the standard deviation?)
You try: What is the range of the set in Example 2? What is the median?
You try: For the data set, find the mean, median and mode. Then give the range and the standard
deviation. (*Rewrite the data from the least to the greatest!)
{100, 89, 112, 104, 96, 106, 93 }
Algebra 2 Section 11-1 page 2
Outliers: Measures of central tendency and spread can give misleading representations of the data set
if the set contains one or more outlier. An outlier is a value that is much less or much greater than
most of the other values in the data set.
Ex. 3: A few scores from a recent quiz are 25, 28, 27, 28, 27.
a)
Find the mean, median, mode, range and standard deviation of the set. (Use your calculator!)
b)
A score on a make-up quiz was 15. Find the new mean, median, mode, range and standard
deviation.
c)
Which measure of central tendency does the outlier affect the most?
The least?
d) What effect does the outlier have on the range and the standard deviation?
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How to find Standard Deviation by hand:
Using the set of values,
1)
2)
3)
4)
5)
6)
Find x-bar (the mean) and n (the # of values)
Subtract x-bar from each value
Square each of those differences
Add those squares
Divide that sum by n - - this is the VARIANCE
Take the square root of that value - - this is the STANDARD DEVIATION, or 
-----------------------------------------------------------------------------------------Standard Deviation on the Calculator
1) Enter the values into List 1 in your calculator (STAT – EDIT – L1)
2) Select STAT – CALC – 1-Var Stats and tell where you put your numbers
(type 2nd, 1 for List 1)
3) The results will give you x-bar and , and if you scroll down you will also see
the value for the median. (You’re on your own to figure out the mode!!)
Algebra 2 Section 11-1 page 3