Download Mean

BUS250
Seminar 4
Key Terms
• Mean: the arithmetic average of a set of data
or sum of the values divided by the number of
values.
• Median: the middle value of a data set when
the values are arranged in order of size.
• Mode: the value or values that occur most
frequently in a data set.
Find the mean of a data set
1. Find the sum of the values.
2. Divide the sum by the total number of values.
Mean =
sum of values
number of values
Try these examples
• Mileage for the new salesperson has been
243, 567, 766, 422 and 352 this week.
What is the average number of miles
traveled?
• 470 miles daily
• Prices from different suppliers of 500 sheets
of copier paper are as follows: $3.99, $4.75,
$3.75 and $4.25. What is the average
price?
• $4.19
7.1.2 Find the Median
• Arrange the values in the data set from smallest
to largest (or largest to smallest) and select the
value in the middle.
• If the number of values is odd, it will be exactly
in the middle.
• If the number of values is even, identify the two
middle values. Add them together and divide by
two.
Here is an example
•
A recent survey of the used car market for the particular
model John was looking for yielded several different
prices: $9,400, $11,200, $5,900, $10,000, $4,700,
$8,900, $7,800 and $9,200. Find the median price.
•
Arrange from highest to lowest:
$11,200, $10,000, $9,400, $9,200, $8,900, $7,800,
$5,900 and $4,700.
•
Calculate the average of the two middle values.
•
(9,200 + 8,900) ÷ 2 = $9,050 or the median price
Look at this example
Class
Interval
75-79
80-84
85-89
90-94
95-99
Total
78 84 95 88 99 92 87 94 90 77
Class
Relative
Frequency
Calculations
Frequency
2
2/10
20%
1
1/10
10%
2
2/10
20%
3
3/10
30%
2
2/10
20%
10
10/10
100%
Here’s an example
Local Daycare Market Share
6%
16%
43%
35%
Teddy Bear
La La Land
Little Gems
Other
Key Terms
• Measures of central tendency: statistical
measurements such as the mean, median
or mode that indicate how data groups
toward the center.
• Measures of variation or dispersion:
statistical measurement such as the range
and standard deviation that indicate how
data is dispersed or spread.
Key Terms
• Range: the difference between the highest
and lowest values in a data set. (also called
the spread)
• Deviation from the mean: the difference
between a value of a data set and the mean.
• Standard variation: a statistical
measurement that shows how data is
spread above and below the mean.
Key Terms
• Variance: a statistical measurement that is the
average of the squared deviations of data from
the mean. The square root of the variance is
the standard deviation.
• Square root: the quotient of number which is
the product of that number multiplied by itself.
The square root of 81 is 9. (9 x 9 = 81)
• Normal distribution: a characteristic of many
data sets that shows that data graphs into a
bell-shaped curve around the mean.
7.3.1 Find the Range in a Data
Set
• Find the highest and lowest values.
• Find the difference between the two.
• Example: The grades on the last exam were
78, 99, 87, 84, 60, 77, 80, 88, 92, and 94.
The highest value is 99.
The lowest value is 60.
The difference or the range is 39.
7.3.2 Find the Standard Deviation
• The deviation from the mean of a data value
is the difference between the value and the
mean.
• Get a clearer picture of the data set by
examining how much each data point
differs or deviates from the mean.
Deviations from the mean
• When the value is smaller than the mean, the
difference is represented by a negative
number indicating it is below or less than the
mean.
• Conversely, if the value is greater than the
mean, the difference is represented by a
positive number indicating it is above or
greater than the mean.
Find the deviation from the
mean
• Find the mean of a set of data.
• Mean = Sum of data values
Number of values
• Find the amount that each data value deviates
or is different from the mean.
• Deviation from the mean = Data value - Mean
Here’s an example
• Data set: 38, 43, 45, 44
• Mean = 42.5
• 1st value: 38 – 42.5 = -4.5 below the mean
• 2nd value: 43 – 42.5 = 0.5 above the mean
• 3rd value: 45 – 42.5 = 2.5 above the mean
• 4th value: 44 – 42.5 = 1.5 above the mean
Find the standard deviation
of a set of data
• A statistical measure called the standard
deviation uses the square of each deviation
from the mean.
• The square of a negative value is always
positive.
• The squared deviations are averaged (mean)
and the result is called the variance.
Find the standard deviation
of a set of data
• The square root is taken of the variance so that
the result can be interpreted within the context
of the problem.
• This formula averages the values by dividing by
one less than the number of values (n-1).
• Several calculations are necessary and are best
organized in a table.
Find the standard deviation
of a set of data
1. Find the mean.
2. Find the deviation of each value from the
mean.
3. Square each deviation.
4. Find the sum of the squared deviations.
5. Divide the sum of the squared deviations by
one less than the number of values in the data
set. This amount is called the variance.
6. Find the standard deviation by taking the
square root of the variance.
Find the standard deviation
Find the standard deviation for the following
data set:
18 22 29 27
Deviation
Squares of
Value
Mean
from Mean
Deviation
18
24
18 – 24 = -6
-6 x -6 = 36
22
24
22 – 24 = -2
-2 x -2 = 4
29
24
29 – 24 = 5
5 x 5 = 25
27
24
27 – 24 = 3
3x 3= 9
Sum of Squared Deviations 74
Find the standard deviation
of a set of data
Variance = sum of squared deviations
n–1
Variance = 74 ÷ 3 = 24.666667
Standard deviation = square root of the variance
Standard deviation = 4.97 rounded