2.10: Chapter 2 Formula Review

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2.2 Measures of the Location of the Data

\(i=\left(\frac{k}{100}\right)(n+1)\)

where \(i\) = the ranking or position of a data value,

\(k\) = the \(k\)th percentile,

\(n\) = total number of data.

Expression for finding the percentile of a data value: \(\left(\frac{x+0.5 y}{n}\right)(100)\)

where \(x\) = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile,

\(y\) = the number of data values equal to the data value for which you want to find the percentile,

\(n\) = total number of data

2.3 Measures of the Center of the Data

\(\mu=\frac{\sum f m}{\sum f}\) Where \(f\) = interval frequencies and \(m\) = interval midpoints.

The arithmetic mean for a sample (denoted by \(\overline{x}\)) is \(\overline{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}\)

The arithmetic mean for a population (denoted by μ) is \({\mu}=\frac{\text { Sum of all values in the population }}{\text { Number of values in the population }}\)

2.7 Measures of the Spread of the Data

\(s_{x}=\sqrt{\frac{\sum f m^{2}}{n}-\overline{x}^{2}} \text { where } \) \(\begin{array}{l}{s_{x}=\text { sample standard deviation }} \\ {\overline{x}=\text { sample mean }}\end{array}\)

Formulas for Sample Standard Deviation \(s=\sqrt{\frac{\Sigma({x}_i-\overline{x})^{2}}{n-1}} \text { or } s=\sqrt{\frac{\Sigma {f}_i({x}_i - \overline{x})^{2}}{n-1}} \)

For the sample standard deviation, the denominator is n - 1, that is the sample size - 1.

Formulas for Population Standard Deviation \({\sigma}=\sqrt{\frac{\Sigma({x}_i - \mu)^{2}}{N}} \text { or } \sigma=\sqrt{\frac{\Sigma {f}_i({x}_i - \mu)^{2}}{N}} \)

For the population standard deviation, the denominator is N, the number of items in the population.