Chebyshev's Theorem

Statistics

1. Objectives: Define the Chevyshev’s Theorem Apply the Chevyshev’s Theorem to raw data Calculate values using Chebyshev’s Theorem

2. C hebyshev's T heorem Discussant: REINA ANTONETTE P. FRANCO MAED- INM

3. Chebyshev's theorem  is used to find the proportion of observations you would expect to find within k standard deviations from the mean.  Chebyshev's Theorem

4. Chebyshev's Theorem 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/ % w/in k st. dev. of mean 2 3 4 5 6 7 K (# of standard deviation) % w/in k st. dev. of mean 2 3 4 5 6 7

6. Compute! K (# of standard deviation) 1-1/ % 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% K (# of standard deviation) % 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: score= mean - standard deviation*k (1 st score– the number lower than the mean) score= mean + standard deviation*k (2 nd 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

9. Therefore;

10. Let’s take a look at this! Using Chebyshev , solve the following problem for a distribution with a mean of 80 and a standard dev. of 10. At least what percentage of values will fall between 60 and 100? At least what percentage of values will fall between 65 and 95 ?

11. 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.

12. 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?

13. 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.

14. 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?

15. 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.

16. Summary Chebyshev’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.

17. I hope you learn something from me.