> ## Documentation Index
> Fetch the complete documentation index at: https://v1-learn.neoartd.my.id/llms.txt
> Use this file to discover all available pages before exploring further.
# Measures of Variation
## Range
The `range` is defined to be the highest value minus the lowest value. The symbol `R` is used for the range.
> `range` didefinisikan sebagai nilai tertinggi dikurangi nilai terendah. Simbol `R` digunakan untuk range.
R = highest value – lowest value.
Extremely large or extremely small data values can drastically affect the range.
> Nilai data yang sangat besar atau sangat kecil dapat mempengaruhi range secara drastis.
## Population Variance
The variance is the average of the squares of the distance each value is from the mean.\
The symbol for the population variance is $\sigma^2$ ($\sigma$ is the Greek lowercase letter sigma)
$\sigma^2 = \dfrac{\Sigma (X-\mu)^2}{N}$, where
* $X$ = is individual value
* $\mu$ = is population mean
* $N$ = is population size
### Example
Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20.\
Find the mean and variance.
The `mean` $\mu$ = (10 + 60 + 50 + 30 + 40 + 20)/6 = 210/6 = 35.
The `variance` $\sigma^2 = \dfrac{\Sigma (X-\mu)^2}{N} = \dfrac{1750}{6}$ = 291.67.
Table
| $X$ | $X-\mu$ | $(X-\mu)^2$ |
| :---: | :-----: | :---------: |
| 10 | -25 | 625 |
| 60 | 25 | 625 |
| 50 | 15 | 225 |
| 30 | -5 | 25 |
| 40 | 5 | 25 |
| 20 | -15 | 225 |
| `210` | | `1750` |
### Population Standard Deviation
The standard deviation, denoted by $\sigma$, The standard deviation is the square root of the variance.
$\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{\Sigma (X-\mu)^2}{N}}$
## Sample Variance
The unbiased estimator of the population variance or the sample variance is a statistic whose value approximates the expected value of a population variance.\
It is denoted by $s^2$, where
$s^2 = \dfrac{\Sigma (X-\overline{\rm X})^2}{n-1}$, and
* $\overline{\rm X}$ = sample mean
* $n$ = sample size
### Example
Find the variance and standard deviation for the following sample: 16, 19, 15, 15, 14.
$\Sigma X$ = (16 + 19 + 15 + 15 + 14) = 79
$\Sigma X^2$ = (16^2 + 19^2 + 15^2 + 15^2 + 14^2) = 1263
$s^2 = \dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}$
$~~~~ = \dfrac{1263 - (79)^2 / 5}{4} = 3.7$
$s = \sqrt{3.7} = 1.9$
### Sample Standard Deviation
The sample standard deviation is the square root of the sample variance.
$s = \sqrt{s^2} = \sqrt{\dfrac{\Sigma (X-\overline{\rm X})^2}{n-1}}$
Shortcut Formula for the Sample Variance and the Standard Deviation
$s^2 = \dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}$, or
$s = \sqrt{\dfrac{\Sigma X^2 - (\Sigma X)^2 / n}{n-1}}$
### Sample Variance for Grouped and Ungrouped Data
For `grouped data`, use the class midpoints for the observed value in the different classes.
For `ungrouped data`, use the same formula with the class midpoints, $X_m$, replaced with the actual observed X value.
The sample variance for `grouped data`:
$s^2 = \dfrac{\Sigma f \bullet X_m^2 - [(\Sigma f \bullet X_m)^2 / n]}{n-1}$
For `ungrouped data`, replace $X_m$ with the observe $X$ value.
#### Sample Variance for Grouped Data - Example
| $X$ | $f$ | $f \bullet X$ | $f \bullet X^2$ |
| :-: | :------: | :-----------------: | :--------------------: |
| 5 | 2 | 10 | 50 |
| 6 | 3 | 18 | 108 |
| 7 | 8 | 56 | 392 |
| 8 | 1 | 8 | 64 |
| 9 | 6 | 54 | 486 |
| 10 | 4 | 40 | 400 |
| | $n = 24$ | $f \bullet X = 186$ | $f \bullet X^2 = 1500$ |
#### Sample Variance for Ungrouped Data - Example
The sample variance and standard deviation:
$s^2 = \dfrac{\Sigma f \bullet X^2 - [(\Sigma f \bullet X)^2 / n]}{n-1}$
$~~~~ = \dfrac{1500 - [(186)^2 / 24]}{23} = 2.54$
$s = \sqrt{2.54} = 1.6$
## Coefficient of Variation
The `coefficient of variation` is defined to be the standard deviation divided by the mean. The result is expressed as a percentage.
> Koefisien variasi didefinisikan sebagai standar deviasi dibagi dengan rata-rata. Hasilnya dinyatakan dalam persentase.
$CVar = \dfrac{s}{\overline{\rm X}} \times 100$ or $CVar = \dfrac{\sigma}{\mu} \times 100$
## Chebyshev's Theorem
The proportion of values from a data set that will fall within k standard deviations of the mean will be at least $1 - 1/k^2$, where k is any number greater than 1.
> Proporsi nilai dari kumpulan data yang akan jatuh dalam k standar deviasi dari rata-rata setidaknya $1 - 1/k^2$, di mana k adalah angka lebih besar dari 1.
For k = 2, 75% of the values will lie within 2 standard deviations of the mean. For k = 3, approximately 89% will lie within 3 standard deviations.
> Untuk k = 2, 75% nilai akan berada dalam 2 standar deviasi dari rata-rata. Untuk k = 3, sekitar 89% akan berada dalam 3 standar deviasi.
## The Empirical (Normal) Rule
For any bell shaped distribution:
> Untuk distribusi berbentuk lonceng:
* Approximately 68% of the data values will fall within one standard deviation of the mean.
> Sekitar 68% nilai data akan berada dalam satu standar deviasi dari rata-rata.
* Approximately 95% will fall within two standard deviations of the mean.
> Sekitar 95% akan berada dalam dua standar deviasi dari rata-rata.
* Approximately 99.7% will fall within three standard deviations of the mean.
> Sekitar 99.7% akan berada dalam tiga standar deviasi dari rata-rata.
