Download Algebra III Lesson 61

Algebra III
Lesson 61
Single-Variable Analysis – The Normal
Distribution – Box-and-Whisker Plots
Single-Variable Analysis
Basic Statistics
Mean - the average of the items in question
Range
- the difference between the highest and the lowest
items in question
New Statistics
Standard Deviation
- it is a way of describing how tightly the items
are ‘packed’, and a way of showing how
representative any given number is from the
whole
Deviation
- the first step in finding standard deviation.
Take each number and subtract the mean from
it. This will show how far an item is from the
mean and if it is higher or lower.
Variance
- the second step. Consider it the ‘squared’ average
of the deviations.
Standard Deviation
- take the square root of the variance.
Example 61.1
The measured distances are 12, 7, 15, and 10. Find the mean and the
standard deviation of the distances.
1st – find mean
mean =
2nd – find deviations
12 + 7 + 15 + 10
4
12 – 11 =
15 – 11 =
= 11
1
7 – 11 =
4
10 – 11 =
2
2
2
2
(
)
(
)
(
)
(
)
1
4
4
1
+
−
+
+
−
3rd – find variance var iance =
4
4th – find std dev
σ = 8.5
= 2.92
-4
-1
= 34/4 = 8.5
The Normal Distribution
Run enough experiment and the results will become a normal distribution.
The graph looks like a symmetric mountain, with the peak right
over the mean of the items. And the sides drop off the same on
both sides of the peak.
Median
Mode
- list the numbers in order and find the middle number (
or the average of the middle 2, if an even amount)
-most commonly occurring result
For normal distributions the mean, median, and mode all
occur at the same spot.
mean,
mode,
median
A normal distribution takes on a ‘bell’ shape, and is
commonly also known as a bell curve.
An interesting fact about bell curves is that 1 std dev either side of the
mean will include about 68% of the results. This is the average zone.
2 std devs includes about 95% of the results. 3 std devs includes
99% of the results.
How sharp or flat the curve is depends upon the size of the std dev.
σ=8
σ=2
Example 61.2
The mean of a distribution that is approximately normal is 100 and
the standard deviation is 5. Use the standard deviation to tell how
the data are distributed with respect to the mean.
1 std dev:
68% of the data
95 Æ 105
100 - 1σ = 95
100 + 1σ = 105
2 std dev:
95% of the data
90 Æ 110
100 - 2σ = 90
100 + 2σ = 110
3 std dev:
99% of the data
100 - 3σ = 85
100 + 3σ = 115
85 Æ 115
Box-and-Whisker Plots
A way of graphically showing the results from an experiment, survey, or
test, etc.
To do a box and whisker plot:
1st – put the numbers in order
2nd – find the median, and put a line, and label it, in the list to mark
the median.
Once the median is known half the numbers are above
the median and half below. Not necessarily evenly spread
out, just split in half.
3rd – find the median for the lower half, this marks the upper edge of
the first quarter of the list. Called the first quartile, Q1.
4th – find the median for the upper half, this marks the upper edge of
the third quarter of the list. Called the third quartile, Q3.
5th – make a picture like the one below showing this data.
min
Q1
med
Q3
max
Stem and leaf plot
1st - Arrange the data in order
2nd – Make a vertical list of the leading digits (make
sure they all stop at the same place value)
3rd – list the remaining digits next to its appropriate leader digit(s) in order.
Sample:
16,22,28,33,35,40,50,120
12
5
4
3
2
1
0
0
0
3,5
2,8
6
Example 61.3
The measurements are 16, 22, 28, 33, 35, 40, 50, and 120. find the mean,
the standard deviation, and make a box-and-whisker plot.
1st – find mean
=
2nd – find deviations
3rd
16 + 22 + 28 + 33 + 35 + 40 + 50 + 120
8
16 – 43 =
28 – 43 =
35 – 43 =
50 – 43 =
= 43
-27 22 – 43 = -21
-15 33 – 43 =
-10
-8
40 – 43 = -3
7
120 – 43 = 77
2
2
2
2
2
2
2
2
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
27
21
15
10
8
3
7
77
−
+
−
+
−
+
−
+
−
+
−
+
+
– find variance =
8
= 7546/8 = 943.25
4th – find std dev
σ = 943.25
= 30.71
1st – put the numbers in order
16,22,28,33,35,40,50,120
2nd – find the median, and put a line, and label it, in the list to mark
the median.
16,22,28,33,35,40,50,120
median = 34
3rd – find the median for the lower half, this marks the upper edge of
the first quarter of the list. Called the first quartile, Q1.
16,22,28,33,35,40,50,120
Q1 = 25
Q3 = 45
4th – find the median for the upper half, this marks the upper edge of
the third quarter of the list. Called the third quartile, Q3.
5th – make a picture like the one below showing this data.
min
16
Q1
25
med
34
Q3
45
max
120
Practice
9,11,12