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The sample mean is a measure of the central tendency of a dataset. It provides an average value, offering insight into where the data roughly clusters. To compute the sample mean, especially with frequency distribution, follow these steps:

First, determine the midpoints for each range in your data. This is done by taking the average of the upper and lower interval bounds. For example, the midpoint for the interval 8.6-12.5 is \( \frac{8.6 + 12.5}{2} = 10.55 \). Repeat for all intervals: 12.6-16.5, 16.6-20.5, 20.6-24.5, and 24.6-28.5.

Next, multiply each midpoint by its respective frequency (number of companies falling in that range). Sum these products together to get a total weighted value.

Finally, divide this total by the total number of companies in your sample (in this case, 50) to find the sample mean. This shows the average profit as a percentage of assets for the companies in the sample.

Sample variance helps measure the dispersion or spread of data points around the mean. High variance suggests a wide spread, while low variance indicates that the data points tend to be close to the mean.

To calculate it, start by finding the difference between each midpoint and the sample mean. Square this difference to remove negative values, which results in \((x_i - \bar{x})^2\) for each midpoint. This step ensures that all deviations contribute positively to the variance.

Then, multiply these squared differences by their respective frequencies to get the sum of squared deviations for each class interval.

• For instance: for a class 10.55, deviation is \((10.55 - 15.59)^2 = 25.6001\) and for frequency 15, it becomes \(15 \times 25.6001 = 384.0015\).

Sum all these values and divide by \(n-1\) (where \(n\) is the sample size) to adjust for sample size, yielding the sample variance.

Frequency distribution shows how data points are distributed across different intervals. It allows you to easily see how often different values or ranges of values occur in a dataset.

In the given exercise, data is grouped into five intervals (or "bins") representing the profit as a percentage of assets across 50 companies. Each interval is accompanied by a frequency, showing how many companies fall within that profit range. For instance:

• The interval 8.6-12.5 has a frequency of 15.
• The interval 12.6-16.5 has a frequency of 20, the highest in this case.
• The lower intervals, such as 24.6-28.5, have fewer frequencies.

This distribution helps identify patterns in the data, such as which range is more common or less common among the companies. It's a crucial step before calculating measures like mean and variance, as it provides detailed insights into how data is grouped.

Standard deviation gives a clear picture of how spread out the values in a data set are. It is the square root of the variance, making it an average distance from the mean.

Since variance is in squared units, standard deviation brings it back to the original unit, making it easier to understand and interpret. For instance, if our variance within the context of companies' profit is \(23.467\), then the standard deviation will be \(\sqrt{23.467} \approx 4.844\).

The standard deviation tells us that, on average, the profit percentages deviate about 4.844% from the mean. This information can be essential for understanding the impression or variability of profits in these Fortune 500 companies. It helps companies assess risk and make more informed financial decisions.