The fraction of any set of numbers lying within k standard deviations of those numbers of the mean of those numbers is at least

${1-\frac{1}{k^2}}$

Where −

- ${k = \frac{the\ within\ number}{the\ standard\ deviation}}$

and ${k}$ must be greater than 1

### Example

**Problem Statement:**

Use Chebyshev’s theorem to find what percent of the values will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.

**Solution:**

- We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.
- We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.
- Those two together tell us that the values between 123 and 179 are all within 28 units of the mean. Therefore the “within number” is 28.
- So we find the number of standard deviations, k, which the “within number”, 28, amounts to by dividing it by the standard deviation:

So now we know that the values between 123 and 179 are all within 28 units of the mean, which is the same as within k=2 standard deviations of the mean. Now, since k > 1 we can use Chebyshev’s formula to find the fraction of

the data that are within k=2 standard deviations of the mean. Substituting k=2 we have:

So ${\frac{3}{4}}$ of the data lie between 123 and 179. And since ${\frac{3}{4} = 75}$% that implies that 75% of the data values are between 123 and 179.

Table of Contents

1.statistics adjusted rsquared

2.statistics analysis of variance

4.statistics arithmetic median

8.statistics best point estimation

9.statistics beta distribution

10.statistics binomial distribution

11.statistics blackscholes model

13.statistics central limit theorem

14.statistics chebyshevs theorem

15.statistics chisquared distribution

16.statistics chi squared table

17.statistics circular permutation

18.statistics cluster sampling

19.statistics cohens kappa coefficient

21.statistics combination with replacement

23.statistics continuous uniform distribution

24.statistics cumulative frequency

25.statistics coefficient of variation

26.statistics correlation coefficient

27.statistics cumulative plots

28.statistics cumulative poisson distribution

30.statistics data collection questionaire designing

31.statistics data collection observation

32.statistics data collection case study method

34.statistics deciles statistics

36.statistics exponential distribution

40.statistics frequency distribution

41.statistics gamma distribution

43.statistics geometric probability distribution

46.statistics gumbel distribution

49.statistics harmonic resonance frequency

51.statistics hypergeometric distribution

52.statistics hypothesis testing

53.statistics interval estimation

54.statistics inverse gamma distribution

55.statistics kolmogorov smirnov test

57.statistics laplace distribution

58.statistics linear regression

59.statistics log gamma distribution

60.statistics logistic regression

63.statistics means difference

64.statistics multinomial distribution

65.statistics negative binomial distribution

66.statistics normal distribution

67.statistics odd and even permutation

68.statistics one proportion z test

69.statistics outlier function

71.statistics permutation with replacement

73.statistics poisson distribution

74.statistics pooled variance r

75.statistics power calculator

77.statistics probability additive theorem

78.statistics probability multiplicative theorem

79.statistics probability bayes theorem

80.statistics probability density function

81.statistics process capability cp amp process performance pp

83.statistics quadratic regression equation

84.statistics qualitative data vs quantitative data

85.statistics quartile deviation

86.statistics range rule of thumb

87.statistics rayleigh distribution

88.statistics regression intercept confidence interval

89.statistics relative standard deviation

90.statistics reliability coefficient

91.statistics required sample size

92.statistics residual analysis

93.statistics residual sum of squares

94.statistics root mean square

96.statistics sampling methods

98.statistics shannon wiener diversity index

99.statistics signal to noise ratio

100.statistics simple random sampling

102.statistics standard deviation

103.statistics standard error se

104.statistics standard normal table

105.statistics statistical significance

108.statistics stem and leaf plot

109.statistics stratified sampling

112.statistics tdistribution table

113.statistics ti 83 exponential regression

114.statistics transformations

116.statistics type i amp ii errors

119.statistics weak law of large numbers