nitraiddQ

2020-10-21

Explain which measure of spread would best describe the payroll:the range,the IQR, or the standard deviation.

### Answer & Explanation

Talisha

Inter quartile range:
In contrast to the range, which measures only differences between the extremes, the inter quartile range (also called mid spread) is the difference between the third quartile and the first quartile. Thus, it measures the variation in the middle 50 percent of the data and unlike the range it is not affected by extreme values. $IQR={Q}_{3}-{Q}_{1}$.
Here, the range and the standard deviation affects by the two higher salaries whereas the interquartile range is not affected by the higher salaries.
Thus, the IQR is the measure of spread that best describe the payroll distribution.

fudzisako

The measure of spread that would best describe the payroll is the \textbf{standard deviation ($\sigma$)}. The standard deviation provides a measure of the average amount by which data points deviate from the mean. It takes into account the entire data set and considers each data point's distance from the mean.
The formula for calculating the standard deviation is as follows:
$\sigma =\sqrt{\frac{1}{N}{\sum }_{i=1}^{N}\left({x}_{i}-\overline{x}{\right)}^{2}}$ where $\sigma$ represents the standard deviation, $N$ is the number of data points, ${x}_{i}$ represents each individual data point, and $\overline{x}$ denotes the mean of the data set.
The range ($R$) would not be the best measure of spread for the payroll because it only considers the difference between the maximum and minimum values and does not take into account the distribution of data points in between.
The interquartile range (IQR) would be a reasonable measure of spread if the data had significant outliers. However, since we are specifically looking for the best measure of spread, the standard deviation, which considers the entire data set, is more appropriate.

xleb123

The choice of measure depends on the nature of the payroll dataset and the specific aspects of spread that need to be emphasized.
Explanation:
$Range$: The range is the difference between the maximum and minimum values in a dataset. It provides a simple measure of the spread, but it can be heavily influenced by extreme values, which may not be representative of the overall distribution. The formula for the range is given as:

$IQR$: The interquartile range is a measure of spread that focuses on the middle 50% of the data. It is less affected by extreme values compared to the range. The IQR is computed as the difference between the upper quartile (Q3) and the lower quartile (Q1). The formula for the IQR is expressed as:
$\text{IQR}=Q3-Q1$
: The standard deviation is a measure that indicates the average amount of variability or dispersion in a dataset. It considers the deviation of each data point from the mean, taking into account the entire distribution. The formula for the standard deviation is given as:

where $N$ is the number of data points, ${x}_{i}$ is each data point, and $\overline{x}$ is the mean of the dataset.
Considering these definitions, the measure of spread that would best describe the payroll depends on the specific characteristics of the dataset.
If the payroll dataset contains extreme values that significantly affect the spread, then the $range$ might be the most appropriate measure. However, if the dataset is less affected by extreme values and there is interest in the dispersion of the middle 50% of the data, then the $IQR$ would be a suitable choice.
On the other hand, if a comprehensive understanding of the overall variability of the payroll is desired, considering all data points, then the would provide a robust measure.

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