Lecture 3

Practical Significance of Standard Deviation

Published

September 5, 2023

2.3 Understanding and Interpreting the Standard Deviation

Given a number k greater than or equal to 1 and a set of n measurements, at least \(1 - \frac{1}{k^2}\) of the measurements will lie within k standard deviations of their mean.
k = 1: 0% of measurments
k = 2: 3/4 of measurements
k = 3: 8/9 of measurements

Interval is computed with \(\mu \pm k\sigma\).

Dotted lines show interval for k = 2

Actual percent of data contained in the interval:

[1] 0.957

Note that it’s much more than the expected 75%

Given \(\bar{x} = 75, s^2 - 100\), create intervals using Tchebysheff’s theorem using k = 1.5, 2, 3
What value of k gives us an interval that contains at least 50% of the measurements.

Another rule works better when data is ‘mound shaped’ or normally distributed.

\(\mu \pm \sigma\) has about 68 % of measurements
\(\mu \pm 2\sigma\) has about 95 % of measurements
\(\mu \pm 3\sigma\) has about 99.7 % of measurements.
We will learn more about where these numbers come from when we learn about the normal distribution.

Using the empirical rule how much of the daa is contained in \(\mu \pm \sigma\)?

First histogram:

[1] 0.6785

Second histogram:

[1] 0.6251598

Sample mean:

[1] 21.6

standard deviation

[1] 5.5

show 1, 2 and 3 intervals of sd from mean and figure out how many datapoints fall in those buckets

k	intervals	freq	rel_freq
1	16.1, 27.1	16	0.64
2	10.6, 32.6	24	0.96
3	5.1, 38.1	25	1.00

Calculate Tchebysheff’s Interval Calculate Empirical Rule Intervals as well

Range for tbl_2_5

[1] 10.2 31.9

[1] 21.7

Std using range approximation

[1] 5.425

Actual Std

[1] 6

[1] "2.2.1, 2.2.3, 2.2.10, 2.3.26"