Transcript AP Stats * 1.1 Overview

AP Stats
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You are going to buy a battery for your video
camera. You have 2 companies to choose from
and they both claim their batteries will last an
average of 40 hours. Both batteries cost the
same. The sample standard deviation of the 2
different batteries is given below. Which
battery would you choose? Why?
Battery A: 1.58
Battery B: 5.3
Section2.4 – Measures of Variation
SWBAT:
Identify and analyze patterns of distributions using shape,
center and spread.
Measures of Variation
Variation is the general term for SPREAD in a data set
Standard Deviation is used to measure spread associated with mean
Deviation is how much a value in a data set differs from the mean
value: SAMPLE: (x- ) or POPULTATION: (x – μ)
Variance is the average squared distance a data value is from the
mean value.
The 5-number summary is used to measure spread associated with the
median.
STANDARD DEVIATION
Std Dev: average distance a value is from the mean
-Measures the spread of the data about the mean
-Enables us to compare the relative spread of the data
with a single number
-Is used with the mean to make inferences about a data
set
-Helps us to make generalization from sets of data
**As with mean, standard deviation is affected by
OUTLIERS and SKEWED DISTRIBUTIONS.**
STANDARD DEVIATION, cont’d
There is a different calculation of standard deviation for
sample and population:
Deviation :
Sample Std Dev (s)
Deviation Squared:
Mean:
Sum of Squares
Steps:
Variance
1) Find the mean of the data set (x-bar)
2) Find the deviation (distance) from the mean of each data item.
3) Square the deviation.
4) Find the sum of squares
5) Divide the sum of squares by (n-1) to get variance (avg. squared
sample deviation) (s2)
6) Take the square root to find the standard deviation (s)
STANDARD DEVIATION, cont’d
Population Std Dev, σ (sigma)
Deviation: (x – μ)
Deviation Squared:
Mean: μ
Sum of Squares
Steps:
1) Find the mean of the data set (mu)
Variance
2) Find the deviation (distance) from the mean of each data item.
3) Square the deviation.
4) Find the sum of squares
5) Divide the sum of squares by N to get variance (avg. squared
deviation) (sigma-squared: σ2)
6) Take the square root to find the standard deviation (σ )
Standard Deviation - Example 1
The means on a 20 point quiz for class A and class B were
both 18. Find the variance, standard deviation, and range
for the two sets of quiz scores.
Class A: 20, 17, 17, 17, 19
Class B: 15, 20, 19, 16, 20
Will you use sample or population standard deviation?
Quiz Scores Mean: 18
x
17
17
17
19
20
Class A: 20, 17, 17, 17, 19
x–μ
17 – 18 = -1
-1
-1
1
2
Variance: σ2 = 1.6
Standard Deviation: σ = √1.6 = 1.26
Range: 20-17 = 3
(x – μ )2
1
1
1
1
4
8
YOU TRY… Mean: 18
x
15
16
19
20
20
Class B: 15, 20, 19, 16, 20
x–μ
15 – 18 = -3
-2
1
2
2
Variance: σ2 = 4.4
Standard Deviation: σ = √4.4 = 2.097
Range: 20-15 = 5
(x – μ )2
9
4
1
4
4
22
Let’s analyze results:
Class A: 20, 17, 17, 17, 19
Variance: σ2 = 1.6
Standard Deviation: σ = √1.6 = 1.26
Range: 20-17 = 3
Class B: 15, 20, 19, 16, 20
Variance: σ2 = 4.4
Standard Deviation: σ = √4.4 = 2.097
Range: 20-15 = 5
Which class scores were more centralized? Which were more spread out?
Is more spread good or bad in this situation?
Can we say that one class did better than the other?
Standard Deviation, cont’d:
Facts about Standard Deviation:
• If Std Dev is small, the data has little spread (ie the
majority of points fall very near the mean).
• As the observations become more spread out about
the mean, the Std Dev increases.
• If Std Dev = 0, there is no spread. This ONLY
happens when ALL data items are the SAME VALUE.
• The Std Dev is significantly affected by outliers and
skewed distributions.
Facts about Standard Deviation, cont’d:
• Std Dev is important to correctly interpret data. For
example, in physical sciences, a lower Std Dev for
the same measurement implies higher precision for
the experiment.
• When interpreting MEAN, you MUST also indicate
the standard deviation.
Our quiz scores from previous slides is an example of why.
Another example: let’s say mean weather over a day in two cities is
240C. However, if the Std Dev is very large, it means likely extremes of
temperatures exist (really hot during day/but cold at night like desert).
On the other hand, if Std Dev is small, it means a fairly uniform temp
throughout day (like coastal region).
Standard Deviation - Example 2
Here are the measurements of the level of phosphate in the
blood of a patient, in milligrams of phosphate per deciliter of
blood, made on six consecutive visits to a clinic.
5.6 5.2 4.6 4.9 5.7 6.4
Compute the mean, variance, and standard deviation. What
does this tell us?
Mean: x-bar = 5.4
Variance: s2 = 0.41
Standard Deviation: s = 0.64
This tells us that overall the readings for THIS PATIENT are fairly close to his average
readings for this group of six. HOWEVER…it doesn’t tell us enough compared to
“good” levels of phosphate. We also don’t know if small changes are in fact
significant or dangerous.
Interpreting Standard Deviation
Without calculating, determine which set has a greatest/lowest standard deviation.
Explain.
HOMEWORK:
P 92. 1, 3, (show work), 6-10all, 14-17all and
worksheet for Std Dev.