Question

Suppose that some measurements occur more than once and that the data \(x_{1}, x_{2}, \ldots, x_{k}\) are arranged in a frequency table as shown here: $$ \begin{array}{cc} \text { Observations } & \text { Frequency } f_{i} \\ \hline x_{1} & f_{1} \\ x_{2} & f_{2} \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ x_{k} & f_{k} \end{array} $$ The formulas for the mean and variance for grouped data are \(\bar{x}=\frac{\sum x_{i} f_{i}}{n}\) $$ \text { where } n=\Sigma f_{i} $$ and $$ s^{2}=\frac{\sum x_{i}^{2} f_{i}-\frac{\left(\sum x_{i} f_{i}\right)^{2}}{n}}{n-1} $$ Notice that if each value occurs once, these formulas reduce to those given in the text. Although these formulas for grouped data are primarily of value when you have a large number of measurements, demonstrate their use for the sample \(1,0,0,1,3,1,3,2,3,0,\) 0,1,1,3,2 a. Calculate \(\bar{x}\) and \(s^{2}\) directly, using the formulas for ungrouped data. b. The frequency table for the \(n=15\) measurements is as follows: $$ \begin{array}{ll} x & f \\ \hline 0 & 4 \\ 1 & 5 \\ 2 & 2 \\ 3 & 4 \end{array} $$ Calculate \(\bar{x}\) and \(s^{2}\) using the formulas for grouped data. Compare with your answers to part a.

Short answer

Answer: The mean and variance for both ungrouped and grouped data are 1.4 and 1.82857142857, respectively. Yes, the results for both methods match, demonstrating the validity of both approaches.

Step-by-step solution

Calculate Mean and Variance Using Ungrouped Data Formulas

First, we will use the formulas for ungrouped data to calculate the mean and variance. The mean for ungrouped data is given by the sum of the measurements divided by the number of measurements, i.e.: $$\bar{x} = \frac{\sum x_i}{n}$$ where \(x_i\) are the measurements and \(n\) is the number of measurements. The variance for ungrouped data is given by the sum of the squared differences between each measurement and the mean, divided by the number of measurements (\(n-1\)), i.e.: $$s^2 = \frac{\sum (x_i -\bar{x})^2}{n-1}$$ Given the sample: \(1,0,0,1,3,1,3,2,3,0, 0,1,1,3,2\). We can calculate the mean: $$\bar{x}=\frac{1+0+0+1+3+1+3+2+3+0+0+1+1+3+2}{15}=\frac{21}{15}=$$ 1.4 Now we can calculate the variance: $$s^2=\frac{(1-1.4)^2+(0-1.4)^2+(0-1.4)^2+(1-1.4)^2+(3-1.4)^2+(1-1.4)^2+(3-1.4)^2+(2-1.4)^2+(3-1.4)^2+(0-1.4)^2+(0-1.4)^2+(1-1.4)^2+(1-1.4)^2+(3-1.4)^2+(2-1.4)^2}{14}=\frac{25.6}{14}=$$ 1.82857142857

Calculate Mean and Variance Using Grouped Data Formulas

Next, we will use the formulas for grouped data to calculate the mean and variance. For the mean: $$\bar{x}=\frac{\sum x_{i} f_{i}}{n}$$ where \(x_i\) are the measurements, \(f_i\) are their respective frequencies, and \(n\) is the total number of measurements. For the variance: $$s^{2}=\frac{\sum x_{i}^{2} f_{i}-\frac{\left(\sum x_{i} f_{i}\right)^{2}}{n}}{n-1}$$ Using the frequency table provided: $$ \begin{array}{ll} x & f \\\ \hline 0 & 4 \\\ 1 & 5 \\\ 2 & 2 \\\ 3 & 4 \end{array} $$ We can calculate the mean: $$\bar{x}=\frac{(0\cdot4)+(1\cdot5)+(2\cdot2)+(3\cdot4)}{15}=\frac{21}{15}=$$ 1.4 And now we can calculate the variance: $$s^2=\frac{(0^2 \cdot 4)+(1^2 \cdot 5)+(2^2 \cdot 2)+(3^2 \cdot 4)-\frac{21^2}{15}}{14}=\frac{56-49}{14}=\frac{25.6}{14}=$$ 1.82857142857

Compare the Results

Now we can compare the results from Step 1 and Step 2. Mean using ungrouped data formula: \(\bar{x}=1.4\) Mean using grouped data formula: \(\bar{x}=1.4\) Variance using ungrouped data formula: \(s^2=1.82857142857\) Variance using grouped data formula: \(s^2=1.82857142857\) The results of the mean and variance from both methods are equal, which demonstrates the validity of both approaches in this given example.

Key concepts

Mean Calculation

The mean, often called the average, is a foundational concept in statistics. It represents the central tendency of a data set. To calculate the mean for any set of data, sum up all the values and then divide by the number of values in the data set. This measure gives us an idea about the "typical" value in the data.
For ungrouped data, the formula for calculating the mean is simple:

• \ (\bar{x} = \frac{\sum x_i}{n})

Where \(x_i\) are the individual data points and \(n\) is the total number of observations. This ensures each data point contributes equally to the final mean.
In the case of grouped data, observations are put in classes or categories with a frequency count for each. The mean formula adapts slightly to accommodate these frequencies:

• \ (\bar{x} = \frac{\sum x_{i} f_{i}}{n})

Here, \(x_i\) are the observed values, \(f_i\) are their respective frequencies, and \(n\) is the total frequency. This approach weights the observations by how often they appear.

• In practical terms, for instance, if you have 3 instances of 5, it means calculating 5 * 3 rather than adding 5 three times.

Variance Calculation

Variance measures how much the values in your dataset deviate from the mean. It's a way to express the spread or variability of the data. The higher the variance, the more spread out the values are. For ungrouped data, you calculate variance as the average of the squared differences between each value and the mean.
The formula is:

• \ (s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1})

Where \(x_i\) are individual data points, \(\bar{x}\) is the mean, and \(n\) is the number of data points. Note the \(n-1\), which provides an 'unbiased' estimate of variance by using the sample size.
With grouped data, variance calculation must account for the frequency of each observation in the dataset:

• \ (s^{2} = \frac{\sum x_{i}^{2} f_{i} - \frac{\left(\sum x_{i} f_{i}\right)^{2}}{n}}{n-1})

Here, \(x_i\) and \(f_i\) are the values and their frequencies, respectively. By considering these factors, grouped variance reflects both how values are averaged and how spread apart they are when weighted by frequency.
Understanding variance is crucial. It aids in determining consistency and reliability across data points.

Ungrouped Data

Ungrouped data refers to data points that are not organized formally beyond their basic arrangement. Each data point stands alone, and there's no categorization. It's raw, individualistic data. Examples include age in years or points on a test.
When calculating statistical measures like mean or variance, ungrouped data uses straightforward arithmetic. Every data point is considered equally.
Understanding how to work with ungrouped data is vital. It allows flexibility in analysis since you can access and modify individual entries directly.
Working with ungrouped data forms the foundation. Before any classification, we assess data through singular observations. This groundwork provides insights into the basics of statistical methods.
When dealing with ungrouped datasets, always remember the sequential nature and variation within the data points. They reveal the precise nature of the distribution.

Grouped Data

Grouped data stems from organizing raw data into summarized frequency distributions. This organization makes analysis more efficient, especially when dealing with large datasets.
By grouping data, sets of numbers are replaced with intervals or categories. Each interval accumulates the frequency of data points it encompasses. Instead of assessing individual data, you evaluate the spread within classes or groups.
This approach aids in visualizing data distributions through histograms or polygon graphs. It shows patterns or trends, like where the bulk of data lies.

• For example, in a class survey, instead of listing individual ages, you may group ages into ranges like 10-14, 15-19, etc. This demonstrates how effectively grouping can summarize data without losing essential information.

Statistical calculations, such as mean and variance, adapt in the context of grouped data by factoring in the frequency \(f_i\) of these intervals. This alters how central tendency and variability are considered.
Understanding and applying grouped data concepts enhances data analysis. This method captures the essence of vast datasets efficiently, making interpretations manageable in reports, studies, and statistical predictions.