### Unit 6 PowerPoint - Dunkerton Community Schools

```HELPFUL VOCABULARY
Normal distribution: a symmetrical bellshaped curve with tails that extend
infinitely in both directions from the
mean of a data set.
Standard deviation: A measure of
spread of a data set equal to the square
root of the sum of the squared variances
divided by the number of data values.
NORMAL DISTRIBUTION
NORMAL DISTRIBUTION
In a normal curve, the right side of the
curve is a mirror image of the left side.
The mean of a normal distribution is the
center or balance point of the two sides
(point where the curve is the highest).
STANDARD DEVIATION
The data spreads from the center or
mean of the curve. This spread is called
deviation.
STANDARD DEVIATION
STANDARD DEVIATION
 Inflection point: A point on a normal curve
where it goes from being concave down to being
concave up. On a normal curve, inflection points
occur at 1 standard deviation from the mean.
 68-95-99.7 Rule: The rule that includes the
percentages of data that are within 1, 2, and 3
standard deviations of the mean of a set of
data.
 Variance: A measure of spread of a data set
equal to the mean of the squared variations of
each data value from the mean of the data set.
STANDARD DEVIATION
 Inflection points: where the graph goes from
concave down to concave up. The points are the
same distance from the mean on both sides of
the curve.
 The inflection points are one standard deviation
away from the mean.
 Approximately 68% of all data will fall within one
standard deviation of the mean.
68-95-99.7 RULE
 68% of all data values should be located within
1 standard deviation of the mean
 95% of all data values should be located within
2 standard deviations of the mean
 99.7% of all data values should be located
(X-µ)
within 3 standard deviations of the mean
CALCULATING STANDARD DEVIATION
 Calculate the standard deviation of the following
numbers, which represent a small population.
2, 7, 5, 6, 4, 2, 6, 3, 6, 9
1. Find the mean of the numbers.
2. Fill out the chart below.
X
(X - µ)
(X - µ)²
CALCULATING STANDARD DEVIATION
Mean= 5.0
Standard deviation Formula:
( − )2

=
X
(X - µ)
(X - µ)²
2
2 – 5 = -3
(-3)² = 9
7
7–5=2
(2)² = 4
5
5–5=0
(0)² = 0
6
6–5=1
(1)² = 1
4
4 – 5 = -1
(-1)² = 1
2
2 – 5 = -3
(-3)² = 9
6
6–5=1
(1)² = 1
3
3 – 5 = -2
(-2)² = 4
6
6–5=1
(1)² = 1
9
9–5=4
(4)² = 16
(9 + 4 + 0 + 1 + 1 + 9 + 1 + 4 + 1 + 16)
10
CALCULATING STANDARD DEVIATION
 Find the standard deviation of the following two
sets of numbers.
 40, 50, 35, 40, 45, 30
 15, 65, 55, 35, 45, 25
 What do you notice about the means of the
data? What do you notice about the standard
variation of the data?
VARIANCE
 Variance of a set of data is the square of the
standard deviation.
 Variance = σ²
 Standard Deviation = σ
NORMAL DISTRIBUTION CURVE
EXAMPLE
 The lifetimes of a certain type of light bulb are
normally distributed. The mean life is 400
hours, and the standard deviation is 75 hours.
For a group of 5,000 light bulbs, how many are
expected to last each of the following times?
 Between 325 hours and 475 hours
 More than 250 hours
 Less than 250 hours
EXAMPLE
 A bag of chips has a mean mass of 70 g, with a
standard deviation of 3 g. Assuming a normal
distribution, create a normal curve including all
necessary values.
 If 1,250 bags of chips are processed each day,
how many bags will have a mass between 67
and 73 g?
 What percentage of the bags of chips will have
a mass greater than 64?
Z-SCORE
A measure of the number of standard deviations
a particular data point is away from the mean
of the sample data.
−
=

EXAMPLE
The grades on a statistics mid-term for a high
school are normally distributed, with a mean
of 81, standard deviation of 6.3. Calculate the
z-scores for each of the following exam grades:
65, 83, 93, 100.
-2.54
.32
1.90
3.02
EXAMPLE
A selective college only admits students who
place at least 2.5 z-scores above the mean on
the ACT with a mean of 19 and a standard
deviation of 5. What is the minimum score
that an applicant must obtain to be admitted
to the university?
PRACTICE
Z-Score Practice Worksheet
- Work on this for the rest of the class period;
due tomorrow.
PERCENTAGES
A z- score chart tells you the percentage of area
BELOW (to the left) of your given z-score.
Need to know the mean and standard deviation.
EXAMPLE
If z = 1, then 84.3% of the data is below that
given value.
If z = 1, then 15.87% of the data is above that
given value.
CHART
Look at the chart given to you.
Find a z-score of -1.45.
What percentage of area is BELOW this z-score?
.0735 or 7.35%
CHART
Look at the chart given to you.
Find a z-score of 3.19.
What percentage of area is BELOW this z-score?
What percentage of area is ABOVE this z-score?
.9993 or 99.93%
.0007 or 0.07%
EXAMPLE
On a nationwide math test, the mean was 65
and the standard deviation was 10. If Robert
scored 81, what was his z-score?
What percentage of students scored below
Robert?
What percentage of students scored above
Robert?
FINDING AREA TO THE RIGHT AND LEFT OF A ZSCORE
LEFT:
Find the z-score on the chart; that is your area.
RIGHT
Find the z-score on the chart; subtract from 1.
FINDING AREA BETWEEN TWO Z-SCORES
Find the area of the whole and subtract the area
of the smaller portion.
Z-SCORES AND PROBABILITY
Find the probability:
P(0 < z < 1.25)
P( -1.3 < z < 2)
.3944 or 39.44%
.8804 or 88.04%
```