Data Dispersion - Range, Variance and Standard Deviation

What is Data Dispersion?

Data dispersion refers to the extent to which data points in a dataset are spread out or scattered. It measures the variability or spread of the data, indicating how much the data points differ from each other and from the mean (average) of the dataset.

Range

The range is a measure of variability that indicates the difference between the highest and lowest values in a dataset. It provides a quick sense of the spread or dispersion within a set of values. While the range is easy to calculate, it is heavily influenced by outliers, which can give a distorted view of the dataset’s spread.

Formula

The range is calculated by subtracting the smallest value in the dataset from the largest value. Mathematically, the range of a dataset with values $x_1, x_2, \ldots, x_n$ is given by:
$R = x_{\text{max}} - x_{\text{min}}$

Where:
$R$ = range
$x_{\text{max}}$ = largest value in the dataset
$x_{\text{min}}$ = smallest value in the dataset

Example Calculation

Consider the following dataset: 3, 7, 8, 5, 10.
To find the range, we follow these steps:

1. Identify the largest and smallest values:
Largest value ($x_{\text{max}}$) = 10
Smallest value ($x_{\text{min}}$) = 3

2. Subtract the smallest value from the largest value:
$R = 10 - 3 = 7$

Therefore, the range of the dataset is 7.

Properties of the Range

1. Simplicity: The range is simple to calculate and understand, making it a straightforward measure of variability.

2. Sensitivity to Outliers: The range is highly sensitive to outliers. A single extreme value can significantly increase the range, which may not accurately represent the overall variability of the dataset.

3. Non-resistance: The range does not resist changes in the dataset. Adding or removing values can greatly affect the range.

4. Limited Information: The range provides only limited information about the dataset’s variability. It does not consider the distribution of values between the extremes.

5. Applicability: The range is applicable to quantitative data where the order of values matters. It is not suitable for categorical data.

Variance

Variance is a measure of dispersion that quantifies the spread or variability of a set of values around their mean. It provides insight into how much individual values deviate from the average, offering a deeper understanding of the dataset's distribution.

Formula

The variance is calculated by averaging the squared differences between each value and the mean. Mathematically, the variance of a dataset with $n$ values $x_1, x_2, \ldots, x_n$ is given by:
$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2$

Where:
$\sigma^2$ = variance
$\mu$ = mean
$n$ = number of values in the dataset
$x_i$ = each individual value in the dataset

Example Calculation

Consider the following dataset: 3, 7, 8, 5, 10 with a mean of 6.6.
To find the variance, we follow these steps:

1. Calculate the squared differences from the mean:
$(3 - 6.6)^2 + (7 - 6.6)^2 + (8 - 6.6)^2 + (5 - 6.6)^2 + (10 - 6.6)^2 = 68.8$

2. Divide by the number of values ($n = 5$):
$\sigma^2 = \frac{68.8}{5} = 13.76$

Therefore, the variance of the dataset is 13.76.

Properties of Variance

1. Non-Negativity: Variance is always non-negative, as it measures the squared deviations from the mean.

2. Sensitivity to Deviations: Variance is sensitive to deviations from the mean, giving more weight to larger deviations compared to smaller ones.

3. Units Squared: Since variance involves squaring the differences, its unit of measurement is squared, which may not be directly interpretable.

4. Importance in Statistical Analysis: Variance is a fundamental measure in statistical analysis, serving as the basis for other statistical techniques such as standard deviation and hypothesis testing.

5. Normalization: Standard deviation, the square root of variance, is often preferred for interpretation as it shares the same unit as the original data and provides a measure of spread comparable to the mean.

Standard Deviation

Standard deviation is a measure of dispersion that quantifies the spread or variability of a set of values around their mean. It is the square root of the variance and provides a more interpretable measure of spread since it is in the same units as the original data.

Formula

The standard deviation is calculated by taking the square root of the variance. Mathematically, the standard deviation of a dataset with $n$ values $x_1, x_2, \ldots, x_n$ is given by:
$\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}$

Where:
$\sigma$ = standard deviation
$\sigma^2$ = variance
$\mu$ = mean
$n$ = number of values in the dataset
$x_i$ = each individual value in the dataset

Example Calculation

Consider the same dataset: 3, 7, 8, 5, 10 with a mean of 6.6 and a variance of 13.76.
To find the standard deviation, we take the square root of the variance:
$\sigma = \sqrt{13.76} \approx 3.71$

Therefore, the standard deviation of the dataset is approximately 3.71.

Properties of Standard Deviation

1. Interpretability: Standard deviation shares the same unit as the original data, making it more interpretable than variance.

2. Sensitivity to Deviations: Like variance, standard deviation is sensitive to deviations from the mean, providing a measure of the spread of data points.

3. Normalization: Standard deviation is often preferred over variance for interpretation as it is in the same units as the original data.

4. Role in Statistical Analysis: Standard deviation is widely used in statistical analysis, particularly in describing the distribution of data and comparing variability between different datasets.

5. Relationship with Variance: Standard deviation is the square root of variance, providing a more intuitive measure of spread that is easier to relate to the original data.

Example Question: A company produces smartphones, and the quality control team monitors the battery life of the smartphones to ensure customer satisfaction. The team collects data on the battery life of a sample of 50 smartphones randomly selected from the production line. The battery life (in hours) of each smartphone in the sample is recorded as follows:

8.1,7.5,8.3,8.0,7.9,8.2,7.8,8.4,7.6,8.1,
8.0,8.2,7.7,8.3,7.9,8.0,8.1,8.4,7.9,8.0,
8.2,8.0,7.8,8.5,7.7,8.1,8.2,8.0,7.9,8.3,
8.1,7.9,8.2,8.3,7.8,8.0,7.7,8.1,8.2,8.0,
8.4,7.6,8.3,8.1,7.9,8.2,7.8,8.0,7.9,8.1

a) Calculate the mean battery life of the sample.
b) Determine the variance and standard deviation of the battery life of the sample.
c) Interpret the results obtained in parts (a) and (b) in the context of the battery life of the smartphones produced by the company. Discuss how the mean, variance, and standard deviation provide insights into the consistency and variability of the battery life across the sample.
Note: Round your answers to two decimal places where applicable.

a) To calculate the mean battery life of the sample, we sum up all the battery life values and divide by the total number of smartphones:

$\text{Mean} = \frac{8.1 + 7.5 + 8.3 + \ldots + 8.0 + 7.9 + 8.1}{50}$

$\text{Mean} = \frac{404.9}{50}$

$\text{Mean} \approx 8.10 \, \text{hours}$

b) To determine the variance and standard deviation, we first need to calculate the deviations of each battery life value from the mean:

Next, we calculate the sum of squared deviations:

$\sum (x - \text{Mean})^2 = 0.00 + 0.36 + 0.04 + \ldots + 0.00 + 0.01$

$\sum (x - \text{Mean})^2 \approx 2.98$

Now, we can calculate the variance using the formula:

$\text{Variance} = \frac{\sum (x - \text{Mean})^2}{\text{Number of Values}}$

$\text{Variance} = \frac{2.98}{50}$

$\text{Variance} \approx 0.06 \, \text{hours}^2$

Finally, we find the standard deviation by taking the square root of the variance:

$\text{Standard Deviation} = \sqrt{\text{Variance}}$

$\text{Standard Deviation} \approx \sqrt{0.06}$

$\text{Standard Deviation} \approx 0.24 \, \text{hours}$

c) The mean battery life of the sample is approximately 8.10 hours. This indicates that, on average, the smartphones produced by the company have a battery life of 8.10 hours.

The variance of approximately 0.06 hours squared and the standard deviation of approximately 0.24 hours provide insights into the variability of the battery life across the sample. A smaller standard deviation suggests that the battery life values are closer to the mean, indicating less variability. Conversely, a larger standard deviation would suggest greater variability in battery life among the smartphones.

In summary, the mean, variance, and standard deviation offer valuable information about the consistency and variability of the battery life of the smartphones produced by the company. These statistics help the quality control team assess the performance and reliability of the smartphones' batteries and make informed decisions to improve product quality.