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
IWBAT summarize data, using measures
of central tendency, such as the mean,
median, mode, and midrange.

Statistic
› A characteristic or
measure obtained
by using the data
values from a
sample

Parameter
› A characteristic or
measure obtained
by using all the data
values from a
specific population

Mean is also known as an arithmetic
average. It is found by adding the values
of the data and dividing by the total
number of values.

Sample Mean

Population Mean

General Rule:
› Wait until the end to round. All calculations in
between should not be rounded.

Rule for Mean:
› The mean should be rounded to one more
decimal place than the raw data.

Using the frequency
distribution, find the mean.
A
Class
B
Frequency
(f)
C
Midpoint
(Xm)
Step 1: Create table
D
(f)(Xm)
Step 2: Find the midpoints
of each class.
5.5-10.5
1
8
8
10.5-15.5
2
13
26
15.5-20.5
3
18
54
20.5-25.5
5
23
115
25.5-30.5
4
28
112
30.5-35.5
3
33
99
35.5-40.5
2
38
76
n = 20
Sum=490
Step 3: For each class,
multiply the frequency by
the midpoint.
Step 4: Find the sum of
column D.
Step 5: divide the sum by
n to get the mean.
The midpoint of the data
 Symbol for median is MD
 To find median

› Step 1: arrange data in order
› Step2 : select the middle point
› If there are 2 middle numbers, add the
numbers and divide by 2.
The number that occurs most often
 Types of modes

›
›
›
›
No mode
Unimodal – one mode
Bimodal – 2 modes
Multimodal – more than 2 modes
› The mode for grouped data is the modal
class. The modal class is the class with the
largest frequency.
The midrange is the sum of the largest
value and the smallest value, divided by
2.
 The symbol for midrange is MR.


To find the weighted mean, multiply
each value by its corresponding weight
and divide the sum of the products by
the sum of the weights.
Course
Credits (w)
Grade (X)
English
3
A (4 points)
Intro to Psychology
3
C (2 points)
Biology I
4
B (3 points)
Physical Education
2
D (1 point)
The grade point average is 2.7.
Measures of
Variation

IWBAT describe data, using measures of
variation, such as the range, variance,
and standard deviation.
The range is the highest number minus
the lowest number. It is represented by R.
 One extremely high or one extremely low
number can affect the range.

The variance is the
average of the squares
of the distance each
value is from the mean.
 Symbol is 2
 Formula is

› 2 = (-)2/
›  is lowercase sigma
›  is the individual value
›  is the mean
›  is the number of data
values
The standard
deviation is the square
root of the variance.
 Symbol is .
 Formula is

›  = 2 = (-)2/
Step 1: Find the mean for the data.
 Step 2: Subtract the mean from each data
value.
 Step 3: Square each result.
 Step 4: Find the sum of the squares.
 Step 5: divide the sum by N to get the
variance.
 Step 6: Take the square root of the
variance to find the standard deviation.

Find the variance and standard deviation of
the following data: 10, 60, 50, 30, 40, 20.
Step 1: Mean is (10+60+50+30+40+20)/6
210/6 = 35
Step 2: Subtract mean from each data value
10 - 35 = -25
60 – 35 = 25
50 – 35 = 15
30 – 35 = -5
40 – 35 = 5
20 – 35 = -15
Step 3: Square each result.
(-25)2 = 625
(25)2 = 625
(15)2 = 225
(-5)2 =25
(5)2 = 25
(-15)2 =225
Step 4: Find the sum of the squares.
625+625+225+225+25+25 = 1750
Step 5: Divide the sum by N to get the variance.
Variance = 1750/6 = 291.7
Step 6: Take the square root to get the
standard deviation.
291.7 = 17.1
Variance and Standard deviation will never be
negative.

Find the Variance and Standard
Deviation of the following data.
35, 45, 30, 35, 40, 25
Variance is 250/6 = 41.7
 Standard deviation = 6.5




When the means are equal, we need to do
more tests to analyze the data. Therefore, we
can determine the variability of the data by
finding the variance and the standard
deviation. Variability is also known as the
spread of the data. The larger the variance
and the standard deviation the more variable
the data is or spread out the data is.
For example, in the manufacture of fittings,
such as nuts and bolts, the variation in
diameter must be small, or parts will not fit
together.
The standard deviation also tells you how far
the data is away from the mean.







Process is similar to finding the mean for grouped
data.
Step 1: Make a table
Step 2: Multiply the midpoint by the frequency.
Step 3: Multiply the frequency by the square of
the midpoint.
Step 4: Find the sum of the frequency, step 2
(m), and step 3 (m2).
Step 5: Find the variance by using this formula
s2 = n( m2 )-( m)2
n(n-1)
Step 6: Take the square root of step 5 to find the
standard deviation.
Class
Frequency
Midpoint (Xm)
m
5.5 – 10.5
1
8
8
64
10.5 – 15.5
2
13
26
338
15.5 – 20.5
3
18
54
972
20.5 – 25.5
5
23
115
2645
25.5 – 30.5
4
28
112
3136
30.5 – 35.5
3
33
99
3267
35.5 – 40.5
2
38
76
2888
n = 20
s2 = n( m2 )-( m)2
n(n-1)
s2 = 20(13310) – (490)2
20(20-1)
s2 = 266200 – 240100
20(19)
s2 = 26100
380
s2 = 68.7
m = 490
s = 68.7 = 8.3
m2
m2 = 13310
If the units of two sets of data are
different we can use the coefficient of
variation to compare the standard
deviations.
 The coefficient of variation is the
standard deviation divided by the mean
and is expressed as a percent.
 Symbol for the coefficient of variation is
CVar.
 Formula is CVar = s/X times 100%

The mean of the number of sales of cars
over a 3-month period is 87, and the
standard deviation is 5. The mean of the
commissions is $5225, and the standard
deviation is $773. Compare the variations of
the two.
 Sales = 5/87 times 100% = 5.7%
 Commissions = 773/5225 times 100% = 14.8%
 The commissions are more variable than the
sales.


The mean for the number of pages of a
sample of women’s fitness magazines is
132, with a variance of 23; the mean for
the number of advertisements of a
sample of women’s fitness magazines is
182, with a variance of 62. Compare the
variations.

The coefficients of variation are
› Pages = 23/132 times 100% = 3.6%
› Advertisements = 62/182 times 100% = 4.3%
› The number of advertisements is more
variable than the number of pages.
Pg.
137
› 1, 2, 16, 18 – 25, 27 – 31

The proportion or percent of values from
a data set that will fall within k standard
deviations of the mean will be at least
1-1/k2, where k is a number greater than
1. (k is also the number of standard
deviations)