3. Chebyshev's theorem is used to find the
proportion of observations you would expect to
find within k standard deviations from the mean.
4. The proportion of any distribution that lies
within k standard deviation of the mean is at
least : , where k is any positive number
greater than 1.
5. Compute!
K (# of standard
deviation)
1-1/𝒌 𝟐 % w/in k st. dev. of
mean
2
3
4
5
6
7
6. Compute!
K (# of standard deviation) 1-1/𝒌 𝟐 % w/in k st. dev. of mean
2 3/4 75%
3 8/9 88.89%
4 15/16 93.75%
5 24/25 96%
6 35/36 97.22%
7 48/49 97.96%
7. Question:
In:
Mean= 80
Standard deviation= 5 (a number used to tell how measurement for a group are
spread out from the average/ mean)
How many percentage of values will it fall between 70 and 90
How are we going to find the “k” standard deviation without
graphing it?
8. Answer:
a. score= mean-standard deviation*k (1st score– the number lower than the
mean)
b. score= mean+standard deviation*k (2nd score– the number w/c is higher
than the mean)
b). 90=80+5K
5k=90-80
5k=10
k=2
a). 70= 80-5k
5k=80-70
5k=10
k=2
10. Let’s take a look at this!
1.Using Chebyshev, solve the following problem for a
distribution with a mean of 80 and a standard dev. of 10.
a. At least what percentage of values will fall between 60
and 100?
a. At least what percentage of values will fall between 65
and 95?
11.
12. Let’s take a look at this!
2. Americans spend an average of 3 hours per
day online. If the standard deviation is 32
minutes, find the range in which at least
88.89% of the data will fall.
13. Let’s activate your !
1. We have 200 data values and the mean is 50 with a
standard deviation of 5, What is the proportion of
values that will fall between the following interval? 30
and 70?
14. Let’s activate your !
2. A professor tells a class that the mean on a
recent exam was 80 with a standard deviation of
6 points, and suppose you wanted to find an
interval where at least 75 percent of the students
must have scored.
15. Let’s activate your !
3. A new college graduate has done their homework and is
searching for their first job. Based on their major, their
educational level, the type of job they are looking for, their
experience, and the geographic location where they want to live,
a salary aggregator tells them that the mean salary of new
employees is approximately 45000 dollars with a standard
deviation of 2600 dollars. This person is subsequently offered a
salary of 52000 dollars. How good is this offer?
16. Facts about Chebyshev
Applicable to any data set – whether it is
symmetric or skewed.
Many times there are more than 75% - this
is a very conservative estimation.
17. Summary
oChebyshev’s Theorem tells us in a rough
abstract way the proportion of data values that
will fall within a certain number of standard
deviations of the mean.
for example, at least 75% of all values of a
distribution fall within two standard
deviations of the mean.
Theorem- a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths. A statement that can be shown to be true.
For example:
Mean= 80
Standard deviation= 5 (a number used to tell how measurement for a group are spread out from the average/ mean)
Range= 70, 90
In chevyshebs theorem, we are going to find the percentage of data would lie within that interval, given the mean and the standard deviation
For any set of data, either population or sample, and for any constant k is greater than 1, the proportion of the data that must lie within k standard deviation on either side of the mean is at least 1-1/k2.
The proportion will never be exact. That’s why we have this word “at least”.
If we set k = 2, then 1 − 1/k2 = 1 − 1/4 = 75%, and Chebyshev’s Theorem tells us that in any data set, at least 75 percent of the data must lie within k = 2 standard deviations of the mean.
If we set k = 3, then 1−1/k2 = 1−1/9 ≈ 88.9%, and Chebyshev’s Theorem tells us that in any data set, at least 88.9 percent of the data must lie within k = 3 standard deviations of the mean.
If we set k = 4, then 1−1/k2 = 1−1/16 = 93.75%, and Chebyshev’s Theorem tells us that in any data set, at least 93.75 percent of the data must lie within k = 4 standard deviations of the mean.
If we set k = 2, then 1 − 1/k2 = 1 − 1/4 = 75%, and Chebyshev’s Theorem tells us that in any data set, at least 75 percent of the data must lie within k = 2 standard deviations of the mean.
If we set k = 3, then 1−1/k2 = 1−1/9 ≈ 88.9%, and Chebyshev’s Theorem tells us that in any data set, at least 88.9 percent of the data must lie within k = 3 standard deviations of the mean.
If we set k = 4, then 1−1/k2 = 1−1/16 = 93.75%, and Chebyshev’s Theorem tells us that in any data set, at least 93.75 percent of the data must lie within k = 4 standard deviations of the mean.
At least 75% of values will fall between 60 and 100
So about 93.7% of the data should live in a our region, but recall how many data we had
2
was 200, so
0.937 ∗ 200 =187.4
So about 187 of our data should live within (30, 70) which was 5 standard deviations from the mean.
Chebyshev's Interval refers to the intervals you want to find when using the theorem. For example, your interval might be from -2 to 2 standard deviations from the mean.
Chebyshev's inequality, also known as Chebyshev's theorem, makes a fairly broad but useful statement about data dispersion for almost any data distribution. This theorem states that no more than 1 / k2 of the distribution's values will be more than k standard deviations away from the mean.
The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
The Empirical Rule is an approximation that applies only to data sets with a bell-shaped relative frequency histogram. It estimates the proportion of the measurements that lie within one, two, and three standard deviations of the mean. Chebyshev's Theorem is a fact that applies to all possible data sets.
