**Quantitative Methods: Basic Concepts**

**Reading 7. Statistical Concepts and Market Returns**

**Learning Outcome Statements**

h. calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean using Chebyshev's inequality;

*CFA Curriculum, 2020, Volume 1*

### Why should I choose AnalystNotes?

Simply put: AnalystNotes offers the **best value**
and the **best product** available to help you pass your exams.

### Subject 7. Chebyshev's Inequality

Let c be any number greater than 1. For any sample or population of data, the proportion of observations that lie FEWER than c standard deviations from the mean is at least:

- This applies to both populations and samples, and to discrete and continuous data, regardless of the shape of the distribution.
- It gives a conservative estimate of the proportion of observations in an interval around the mean.
- It highlights the importance of σ and places lower and upper limits.
- Empirical rule: 68% within 1σ, 95% within 2σ, and 99% within 3σ.

Chebyshev's Inequality theorem is useful in that if we know the standard deviation, we can use it to measure the

*minimum*amount of dispersion.

*Example*

What approximate percent of a distribution will lie within ± two standard deviations of the mean?

Chebyshev's Inequality: 1-(1/c

^{2}) = 1 - (1/2

^{2}) = 0.75 or 75%

Several other values are also useful to know.

- A minimum of 36% of observations lie within 1.25 σ of the mean.
- A minimum of 56% of observations lie within 1.5 σ of the mean.
- A minimum of 75% of observations lie within 2 σ of the mean.
- A minimum of 84% of observations lie within 2.5 σ of the mean.
- A minimum of 89% of observations lie within 3 σ of the mean.
- A minimum of 94% of observations lie within 4 σ of the mean.

###
**User Contributed Comments**
10

You need to log in first to add your comment. ###### sarath

Important concept.

###### itconcepts

why does the empirical rule differ from the "other useful values to know"

###### achu

empirical rule shows the dispersion of a normal distribution (within 1 sd =68%, 2sds 95%). As you can see, the normal distribution dispersion is 'tighter' than that shown by Cheby's inequality, but this is fine. At least 75% must be within 2sd's, but in a normal dist it goes up to 95%.

###### Cesarjr20

What would be a good book to read about this topic?

###### timmak

I would say any entry level university textbook on stats and probability should cover this topic in more detail.

###### Nuta

You can find good practical examples for understanding if you refer to 6 sigma rule.

###### Sophorior

Empirical rule: 68% within 1 sd;, 95% within 2 sd;, and 99% within 3 sd. this is valid for a normal distribution, not for any distribution (so it shoudn't be mentioned here...)

###### tschorsch

is this not a limit of maximum dispersion?

i.e. regardless of the distribution, no more than 1/k² of the values can be more than k std deviations from the mean

###### Ryan1234

That is correct.

###### mydogo

I found this to a much simpler explanation to remember and apply, thus good for retention.