Basic Statistics

Basic statistics is a tool that includes fundamental techniques used to summarize and interpret a given dataset. It encompasses descriptive statistics, such as mean, median, mode, variance, and standard deviation, which help characterize the central tendency and variability of data sets. It also includes confidence intervals, providing a range within which population parameters are likely to fall, and hypothesis testing, which allows analysts to make inferences or decisions based on sample data. Additionally, weighting cases can be applied to adjust for the relative importance of different data points. These tools are essential for understanding data behavior and supporting informed decision-making in research and analysis.

Descriptive Statistics
Confidence Intervals
Hypothesis Testing
Weight Cases
References
Version History

Descriptive Statistics

Descriptive statistics is a collection of analytical methods used to effectively summarize or describe a given dataset. These methods involve various measures, including measures of central tendency (e.g., mean, median, and mode), as well as measures of variability (e.g., range, variance, and standard deviation). By employing descriptive statistics, one can quickly evaluate the distribution and patterns within a dataset without relying on additional assumptions. Descriptive statistics play a crucial role as they offer valuable insights about a dataset in a straightforward manner, without the necessity of graphical representation, which might not have been feasible when dealing with large datasets.¹.

The available statistical measures in Isalos, applicable to both the population and to a sample of the population, include:

Minimum

The minimum (min) is the smallest value in a dataset, defining the lower boundary of the data range. There is no distinction in calculation between a sample and a population for this measure.

Maximum

The maximum (max) is the largest value in a dataset, defining the upper boundary of the data range. Like the minimum, the calculation does not vary between a sample and a population.

Range

The range is a measure of the spread between the maximum and minimum values in the dataset, calculated as:

$$ \begin{equation} R = x_{\text{max}} - x_{\text{min}} {\qquad [1] \qquad} \end{equation} $$

This measure does not differ between population and sample data.

Size/Count

The size or count is the total number of data points in the dataset, denoted as $n$. It is used as the denominator in many statistical formulas to calculate averages and variations. Both population and sample datasets use the same method for counting data points.

Sum

The sum is the total of all numerical values in a dataset, calculated as:

$$ \begin{equation} \sum_{i=1}^{n} x_{i} {\qquad [2] \qquad} \end{equation} $$

Here, $x_{i}$ represents each value in the dataset. The calculation is identical whether dealing with a sample or a population.

Median

The median is the middle value of a dataset when ordered. The method does not vary between sample and population datasets.

If $n$ is odd, the median is the middle number calculated at position $p$ (where $p = \frac{n + 1}{2}$):

$$ \begin{equation} \tilde{x} = x_{p} {\qquad [3a] \qquad} \end{equation} $$

If $n$ is even, it is the average of the two middle numbers at positions $p$ and $p+1$ (where $p = \frac{n}{2}$):

$$ \begin{equation} \tilde{x} = \frac{x_{p} + x_{p+1}}{2} {\qquad [3b] \qquad} \end{equation} $$

Mode

The mode is the value that appears most frequently in a dataset. A dataset may have one mode, multiple modes, or no mode if no number repeats more frequently than others. This calculation is consistent for both population and sample data.

Midrange

The midrange (MR) is the average of the maximum and minimum values of the dataset, calculated as:

$$ \begin{equation} MR = \frac{x_{\text{min}} + x_{\text{max}}}{2} {\qquad [4] \qquad} \end{equation} $$

This calculation is the same for both sample and population data.

Root Mean Square

The root mean square (RMS) provides a measure of the magnitude of a set of numbers, typically used to calculate the standard deviation. It is calculated by taking the square root of the average of the squares of the numbers:

$$ \begin{equation} RMS = \sqrt{\frac{1}{n} \sum_{i=1}^{n} x_{i}^{2}} {\qquad [5] \qquad} \end{equation} $$

This formula is consistent for both population and sample datasets.

Mean

The mean, often referred to as the average, is a measure of central tendency that sums all the numerical values in a dataset and divides by the count of the values. The equation for calculating the mean is:

For populations:

$$ \begin{equation} \mu = \frac{1}{n} \sum_{i=1}^{n} x_{i} {\qquad [6a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} \overline{x} = \frac{1}{n} \sum_{i=1}^{n} x_{i} {\qquad [6b] \qquad} \end{equation} $$

Mean Absolute Deviation

The mean absolute deviation (MAD) is the average of absolute deviations from the mean, providing a measure of variability.

For populations:

$$ \begin{equation} MAD = \frac{1}{n} \sum_{i=1}^{n} |x_{i} - \mu| {\qquad [7a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} MAD = \frac{1}{n} \sum_{i=1}^{n} |x_{i} - \overline{x}| {\qquad [7b] \qquad} \end{equation} $$

Standard Deviation

Standard deviation measures the spread of data around the mean.

For populations:

$$ \begin{equation} \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_{i} - \mu)^{2}} {\qquad [8a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_{i} - \overline{x})^{2}} {\qquad [8b] \qquad} \end{equation} $$

The difference lies in the denominator, where $n-1$ (Bessel’s correction) is used for samples to provide an unbiased estimator.

Variance

Variance measures the average squared deviations from the mean.

For populations:

$$ \begin{equation} \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_{i} - \mu)^{2} {\qquad [9a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_{i} - \overline{x})^{2} {\qquad [9b] \qquad} \end{equation} $$

The distinction is similar to that of standard deviation.

Quartiles

Quartiles divide the data into four equal parts. $Q1$ is the median of the lower half, $Q2$ is the median of the dataset, and $Q3$ is the median of the upper half. The method for calculating quartiles is consistent across sample and population data.

Interquartile Range

The interquartile Range (IQR) is the difference between the third and first quartiles ($Q3$ - $Q1$), measuring the middle 50% spread of the data. There is no variation in calculating IQR between samples and populations.

Outliers

Outliers are data points significantly distant from other observations. They are typically identified using the IQR; values below $Q1 - 1.5 \times IQR$ or above $Q3 + 1.5 \times IQR$ are considered outliers. This identification method applies equally to sample and population data.

Sum of Squares

The sum of squares (SS) is the sum of squared differences from the mean.

For populations:

$$ \begin{equation} SS = \sum_{i=1}^{n} (x_{i} - \mu)^{2} {\qquad [10a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} SS = \sum_{i=1}^{n} (x_{i} - \overline{x})^{2} {\qquad [10b] \qquad} \end{equation} $$

This foundational calculation underpins variance and standard deviation.

Standard Error of the Mean

The standard error of the mean (SEM) measures the accuracy with which a sample represents a population.

For populations:

$$ \begin{equation} SEM = \frac{\sigma}{\sqrt{n}} {\qquad [11a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} SEM = \frac{s}{\sqrt{n}} {\qquad [11b] \qquad} \end{equation} $$

Skewness

Skewness ($\gamma_1$) is a measure of the asymmetry of the probability distribution of a real-valued random variable. Positive skewness indicates a distribution with an asymmetric tail extending towards more positive values, and negative skewness indicates a tail that extends towards more negative values.

For populations:

$$ \begin{equation} \gamma_{1} = \frac{ \sum_{i=1}^{n} (x_{i} - \mu)^{3}}{n\sigma^{3}} {\qquad [12a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} \gamma_{1} = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left( \frac{x_{i} - \overline{x}}{s} \right)^{3} {\qquad [12b] \qquad} \end{equation} $$

Kurtosis

Kurtosis ($\beta_2$) measures the “tailedness” of the probability distribution of a real-valued random variable. High kurtosis in a data set is an indicator of substantial outliers.

For populations:

$$ \begin{equation} \beta_{2} = \frac{ \sum_{i=1}^{n} (x_{i} - \mu)^{4}}{n\sigma^{4}} {\qquad [13a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} \beta_{2} = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{4}}{s^{4}} {\qquad [13b] \qquad} \end{equation} $$

Kurtosis Excess

Excess kurtosis ($\alpha_4$) is calculated by subtracting 3 from the standard kurtosis measurement. This adjustment helps to compare the tails of the distribution to those of a normal distribution, which has a kurtosis of 3.

For populations:

$$ \begin{equation} \alpha_{4} = \frac{ \sum_{i=1}^{n} (x_{i} - \mu)^{4}}{n\sigma^{4}} - 3 {\qquad [14a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} \alpha_{4} = \frac{n(n+1)}{(n-1)(n-2)(n-3)} \frac{\sum_{i=1}^{n} (x_{i} - \overline{x})^{4}}{s^{4}} - \frac{3(n-1)^{2}}{(n-2)(n-3)} {\qquad [14b] \qquad} \end{equation} $$

Coefficient of Variation

The coefficient of variation (CV) is a standardized measure of dispersion of a probability distribution or frequency distribution.

For populations:

$$ \begin{equation} CV = \frac{\sigma}{\mu} {\qquad [15a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} CV = \frac{s}{\overline{x}} {\qquad [15b] \qquad} \end{equation} $$

This measure is useful because it allows comparison between distributions with different units or means.

Relative Standard Deviation

The relative standard deviation (RSD) is expressed as a percentage and describes the dispersion or variability relative to the mean of the data.

For populations:

$$ \begin{equation} RSD = \left[ \frac{100 \times \sigma}{\mu} \right]\% {\qquad [16a] \qquad} \end{equation} $$

For samples:

$$ \begin{equation} RSD = \left[ \frac{100 \times s}{\overline{x}} \right]\% {\qquad [16b] \qquad} \end{equation} $$

Frequency

Frequency ($f_1$) refers to the number of times a particular data point or value appears in a dataset. It is a simple yet fundamental measure used to describe the distribution of data values. The basic formula for frequency is:

$$ \begin{equation} f_{i} = \text{Number of occurrences of value } \times x_{i} {\qquad [17] \qquad} \end{equation} $$

where $f_{i}$ represents the frequency of the $i$-th value $x_{i}$.

Use the Descriptive Statistics function by browsing in the top ribbon:

Statistics $\rightarrow$ Basic Statistics $\rightarrow$ Descriptive Statistics

Input

The input spreadsheet consists of multiple columns, each representing a distinct dataset on which descriptive statistics will be utilized.

Configuration

List of statistics	Select the appropriate methods for conducting descriptive statistics analysis by clicking on the corresponding radio buttons. The available statistics are described above.
Select Type	Select from the dropdown list the option that describes wether the data represents a sample or population.

Output

In the output sheet, descriptive statistics analysis is performed by applying selected statistical measures to each dataset represented in the input sheet. The results display the statistical measures alongside their corresponding datasets in the respective columns.

Example

Input

The input sheet on the left hand-side is organized with individual columns representing each data set. Insert the data sets in a manner that allows for independent analysis of each set when performing descriptive statistics.

Configuration

Select Statistics $\rightarrow$ Basic Statistics $\rightarrow$ Descriptive Statistics.
Click on the statistics that you wish to calculate for the input data [1] and determine whether the data correspond to Population or Sample [2].
Click on the Execute button [3] to calculate the selected statistics.

Output

The right-hand side of the output sheet displays individual datasets per column, accompanied by the corresponding statistical measure chosen on each row.

Confidence Intervals

A confidence interval (CI) is a statistical tool used to estimate the range within which a population parameter (like a mean or proportion) is likely to fall, based on information from sample data¹. This interval is constructed around the sample estimate and provides an indication of where the true parameter might be with a certain level of confidence, usually expressed as a percentage like 95% or 99%.²

To calculate a confidence interval, you typically need the sample mean (which provides a central estimate), the standard deviation (which measures the spread of the data), and the sample size (the number of data points). For proportions, you use the sample proportion instead of the mean. The formula for a confidence interval generally involves the sample statistic plus or minus a margin of error (Eq. 1), which is derived from the standard deviation and the sample size, adjusted by a factor from a statistical distribution (like the Z-score in large samples or the t-score in smaller samples)^1,2.

The confidence interval when considering sample data is calculated based on below formula:

$$ \begin{equation} CI = \left( \overline{x} \pm t_{\text{critical}} \times \frac{s}{\sqrt{n}} \right) {\qquad [1] \qquad} \end{equation} $$

where $\begin{equation} \overline{x} \end{equation}$ is the sample mean, $s$ is the standard deviation of the sample data, $n$ is the sample data size and $\begin{equation}t_{\text{critical}} \end{equation}$ is derived from the t-distribution, which accounts for degrees of freedom (df) that depend on the sample size and is found using Eq. 2 when confidence level is for example 95%:

$$ \begin{equation} t_{\text{critical}} = t_{\frac{\alpha}{2}, df} {\qquad [2] \qquad} \end{equation} $$

where $\begin{equation}df=n-1\end{equation}$ and $\alpha$ is the significance level which is calculated as $\begin{equation}\alpha=1-\text{confidence level}\end{equation}$. The value $\begin{equation}\frac{\alpha}{2}\end{equation}$ reflects the two-tailed nature of the typical confidence interval (e.g. for 95 % confidence interval, $\begin{equation}\alpha=0.05\end{equation}$ and $\begin{equation}\frac{\alpha}{2}=0.025\end{equation}$).²

The confidence interval when considering population data is calculated based on Eq. 3:

$$ \begin{equation} CI = \left( \mu \pm Z_{\text{critical}} \times \frac{\sigma}{\sqrt{n}} \right) {\qquad [3] \qquad} \end{equation} $$

Where $μ$ is the population mean, $σ$ is the standard deviation of the population data, $n$ is the population size and $\begin{equation}Z_{\text{critical}}\end{equation}$ is derived from the standard normal distribution (Z-distribution), which is symmetric and has a mean of zero and a standard deviation of one. The $\begin{equation}Z_{\text{critical}}\end{equation}$ value for a given confidence level is:

$$ \begin{equation} Z_{\text{critical}} = Z_{\frac{\alpha}{2}} {\qquad [4] \qquad} \end{equation} $$

Where $\frac{\alpha}{2}$ again accounts for the two-tailed nature of the test or interval. For a 95 % confidence interval, you look up or calculate the Z-value that corresponds to 97.5 % of the distribution (since 2.5 % is in each tail).²

The confidence interval when considering population proportion is calculated based on Eq. 5:

$$ \begin{equation} CI = \left( p \pm z_{\text{critical}} \times \sqrt{\frac{p(1-p)}{n}} \right) {\qquad [5] \qquad} \end{equation} $$

where $p$ is the sample proportion and is calculated as $\begin{equation}p = \frac{X}{n}\end{equation}$: where X is the number of successes in the dataset and n is the total sample size.

Isalos offers three methods for calculating confidence intervals. The first method involves inputting raw (sample) data, assumed to be a representative sample. In this method, the mean can be automatically calculated, and the user has the option to manually input the standard deviation value or allow the software to calculate it automatically. This flexible approach allows for accurate estimation of confidence intervals based on the provided sample raw data. The second method requires the user to manually enter the mean value, standard deviation, and sample size. Additionally, the user must indicate whether the data represents a sample or population. The third method calculates confidence intervals based on proportions. Here, the user needs to specify the probability and sample size.

These different calculation options in Isalos provide flexibility and accommodate various scenarios for estimating confidence intervals with ease and accuracy. All the above three options allow the user to manually specify the confidence level.

Raw Data

As previously mentioned, the confidence intervals can be calculated using input raw (sample) data. The mean is automatically calculated, and the user has the option to manually enter or automatically calculate the standard deviation of the sample data. Additionally, the user can specify the desired confidence level for the calculation of the confidence intervals.

Use the Raw Data function for calculating confidence intervals by browsing in the top ribbon:

Statistics $\rightarrow$ Basic Statistics $\rightarrow$ Confidence Intervals $\rightarrow$ Raw Data