Coefficient of quartile variation *(\(CQV\))* is a measure of relative dispersion that is based on interquartile range (`IQR`

). Since *cqv* is unitless, it is useful for comparison of variables with different units. It is also a measure of homogeneity (Altunkaynak & Gamgam, 2018; Bonett, 2006).

The *population* coefficient of quartile variation is:

\[
CQV = \biggl(\frac{Q_3-Q_1}{Q_3+Q_1}\biggr)\times100
\] where \(q_3\) and \(q_1\) are the population third quartile (*i.e.,* \(75^{th}\) percentile) and first quartile (*i.e.,* \(25^{th}\) percentile), respectively. Almost always, we analyze data from samples but want to generalize it as the population’s parameter (Albatineh, Kibria, Wilcox, & Zogheib, 2014). Its sample’s estimate is given as:

\[
cqv = \biggl(\frac{q_3-q_1}{q_3+q_1}\biggr)\times100
\] There are different methods for the calculation of **confidence intervals (CI)** for *CQV*. All of them are fruitful and have particular use cases. For sake of versatility, we cover almost all of these methods in `cvcqv`

package. Here, we explain them along with some examples:

Bonett (Bonett, 2006) introduced the following confidence interval for *CQV*:

\[
\exp\{\ln{(D/S)c\ \pm\ Z_{1-\alpha/2}\sqrt{v} }\}
\] where \(c = n/(n-1)\) is a centering adjustment which helps to equalize the tail error probabilities (Altunkaynak & Gamgam, 2018; Bonett, 2006). \(D = \hat{Q3}-\hat{Q1}\) and \(S = \hat{Q3}+\hat{Q1}\) are the sample \(25^{th}\) and \(75^{th}\) percentiles, respectively; \(Z_{1-\alpha/2}\) is the \(1-\alpha/2\) quantile of the standard normal distribution. Computation of \(v\) which is \(Var\{\ln{(D/S)}\}\) is long and a bit complicated, but has been implemented for `cqv`

function:

```
x <- c(
0.2, 0.5, 1.1, 1.4, 1.8, 2.3, 2.5, 2.7, 3.5, 4.4,
4.6, 5.4, 5.4, 5.7, 5.8, 5.9, 6.0, 6.6, 7.1, 7.9
)
cqv_versatile(
x,
na.rm = TRUE,
digits = 3,
method = "bonett"
)
```

```
## $method
## [1] "cqv with Bonett 95% CI"
##
## $statistics
## est lower upper
## 45.625 24.785 77.329
```

Thanks to package `boot`

(Canty & Ripley, 2017) we can obtain bootstrap CI around \(cqv\):

```
## $method
## [1] "cqv with adjusted bootstrap percentile (BCa) 95% CI"
##
## $statistics
## est lower upper
## 45.625 21.527 79.698
```

In conclusion, we can observe CIs calculated by all available methods:

```
## $method
## [1] "All methods"
##
## $statistics
## est lower upper
## bonett 45.625 24.785 77.329
## norm 45.625 19.410 70.462
## basic 45.625 19.398 73.303
## percent 45.625 17.947 71.852
## bca 45.625 25.532 83.405
## description
## bonett cqv with Bonett 95% CI
## norm cqv with normal approximation 95% CI
## basic cqv with basic bootstrap 95% CI
## percent cqv with bootstrap percentile 95% CI
## bca cqv with adjusted bootstrap percentile (BCa) 95% CI
```

Albatineh, A. N., Kibria, B. M., Wilcox, M. L., & Zogheib, B. (2014). Confidence interval estimation for the population coefficient of variation using ranked set sampling: A simulation study. *Journal of Applied Statistics*, *41*(4), 733–751. https://doi.org/10.1080/02664763.2013.847405

Altunkaynak, B., & Gamgam, H. (2018). Bootstrap confidence intervals for the coefficient of quartile variation. *Communications in Statistics: Simulation and Computation*, *0*(0), 1–9. https://doi.org/10.1080/03610918.2018.1435800

Bonett, D. G. (2006). Confidence interval for a coefficient of quartile variation. *Computational Statistics and Data Analysis*, *50*(11), 2953–2957. https://doi.org/10.1016/j.csda.2005.05.007

Canty, A., & Ripley, B. (2017). boot: Bootstrap R (S-Plus) Functions. R package version 1.3-20.