2.12: Formula Review

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2.3 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 kth 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

2.4 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 $$\bar{x}$$ ) is $$\bar{x}=\frac{\text { Sum of all values in the sample }}{\text { Number of values in the sample }}$$

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

2.6 Geometric Mean

The Geometric Mean: $$\tilde{x}=\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}=\sqrt[n]{x_1 \cdot x_2 \cdots x_n}=\left(x_1 \cdot x_2 \cdots x_n\right)^{\frac{1}{n}}$$

2.7 Skewness and the Mean, Median, and Mode

Formula for skewness: $$a_3=\sum \frac{\left(x_i-\bar{x}\right)^3}{n s^3}$$
Formula for Coefficient of Variation: $$C V=\frac{s}{\bar{x}} \cdot 100$$ conditioned upon $$\bar{x} \neq 0$$

2.8 Measures of the Spread of the Data

$$s_x=\sqrt{\frac{\sum f m^2}{n}-\bar{x}^2}$$ where $$\begin{array}{l}s_x=\text { sample standard deviation } \\ \bar{x}=\text { sample mean }\end{array}$$
$$\bar{x}=$$ sample mean

Formulas for Sample Standard Deviation $$s=\sqrt{\frac{\Sigma(x-\bar{x})^2}{n-1}}$$ or $$s=\sqrt{\frac{\Sigma f(x-\bar{x})^2}{n-1}}$$ or $$s=\sqrt{\frac{\left(\sum_{i=1}^n x^2\right)-n \bar{x}^2}{n-1}}$$ For the sample standard deviation, the denominator is $$\boldsymbol{n} \mathbf{- 1}$$, that is the sample size - 1 .

Formulas for Population Standard Deviation $$\sigma=\sqrt{\frac{\Sigma(x-\mu)^2}{N}}$$ or $$\sigma=\sqrt{\frac{\Sigma f(x-\mu)^2}{N}}$$ or $$\sigma=\sqrt{\frac{\sum_{i=1}^N x_i^2}{N}-\mu^2 \text { For }}$$ the population standard deviation, the denominator is $$N$$, the number of items in the population.

2.12: Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.