ringearV

2021-10-17

Prove the following
$\begin{array}{|ccc|}\hline \text{Deviation boundary}& \mathrm{%}\text{of data points within boundary}& \text{Probability of any particular data point being outside boundary}\\ ±\sigma & 68.0& 32.0\mathrm{%}\\ ±2\sigma & 95.40& 4.60\mathrm{%}\\ ±3\sigma & 99.70& 0.30\mathrm{%}\\ \hline\end{array}$

stuth1

Step 1
Empirical Rule :
The empirical rule is used for forecasting final outcomes. After calculating the standard deviation and before collecting exact data, this rule can be used as a rough estimate of the outcome of the impending data to be collected and analyzed.
Approximately $68\mathrm{%}$ of the data lie within one standard deviation of the mean, that is, in the interval with endpoints $x±s$ for samples and with endpoints $\mu ±\sigma$ for populations; if a data set has an approximately bell-shaped relative frequency histogram, then approximately $95\mathrm{%}$ of the data lie within two standard deviations of the mean, that is, in the interval with endpoints $x±2s$ for samples and with endpoints $\mu ±2\sigma$ for populations; and
approximately $99.7\mathrm{%}$ of the data lies within three standard deviations of the mean, that is, in the interval with endpoints $x±3s$ for samples and with endpoints $mi±3\sigma$ for populations.
Step 2

One of the most necessary and sufficient condition for Empirical Rule are that the data distribution must be approximately bell-shaped and that the percentages are only approximately true.
The Empirical Rule does not apply to data sets with severely asymmetric distributions, and the actual percentage of observations in any of the intervals specified by the rule could be either greater or less than those given in the rule.

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