Descriptive Statistics Using StatCrunch: Five-Number Summary, Mean, Variance, and Standard Deviation

Describing Data Numerically

Introduction to Descriptive Statistics

Descriptive statistics are essential tools in Statistics for Business, allowing us to summarize and interpret data sets efficiently. This guide demonstrates how to use StatCrunch to compute key numerical summaries, including the five-number summary, mean, variance, standard deviation, interquartile range (IQR), and mode.

Five-Number Summary

The five-number summary provides a quick overview of the distribution of a data set. It consists of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.

• Minimum: The smallest data value.

• Q1 (First Quartile): The value below which 25% of the data fall.

• Median: The middle value when data are ordered.

• Q3 (Third Quartile): The value below which 75% of the data fall.

• Maximum: The largest data value.

Example: For the data set [56, 67, 68, 72, 74, 75, 88, 90, 97, 99], the five-number summary is:

• Min = 56

• Q1 = 68

• Median = 74.5

• Q3 = 90

• Max = 99

Interquartile Range (IQR)

The interquartile range (IQR) measures the spread of the middle 50% of the data and is calculated as:

$IQR = Q3 - Q1$

Example: For the above data, $IQR = 90 - 68 = 22$

Mean

The mean (average) is a measure of central tendency, calculated by summing all data values and dividing by the number of values:

$\text{Mean} = \frac{\sum x_i}{n}$

Example: For the data set, $\text{Mean} = 76.1$

Variance and Standard Deviation

Variance measures the average squared deviation from the mean, while standard deviation is the square root of the variance, representing the typical distance of data values from the mean.

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

• Sample Standard Deviation: $s = \sqrt{s^2}$

Example:

• Sample variance: $200.93133$

• Sample standard deviation: $14.275096$

Mode

The mode is the value that appears most frequently in a data set. If no value repeats, the data set is said to have "no mode."

Example: For the given data, there is no mode.

Using StatCrunch for Descriptive Statistics

StatCrunch is a statistical software tool that simplifies the calculation of descriptive statistics. The steps to compute these statistics are:

1. Enter the data into StatCrunch, one value per row.

2. From the top menu, select Stat → Summary Stats → Columns.

3. Select the column containing your data.

4. In the Statistics box, hold Ctrl and select multiple statistics (Mean, Variance, Std. dev, Median, Range, Min, Max, Q1, Q3, IQR, Mode).

5. Click Compute to view the results.

Results will display all selected statistics, allowing for quick analysis and interpretation.

Summary Table: Key Descriptive Statistics

Statistic	Definition	Formula (LaTeX)	Example Value
Mean	Average value	$\frac{\sum x_i}{n}$	76.1
Variance	Average squared deviation from mean	$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$	200.93133
Standard Deviation	Square root of variance	$s = \sqrt{s^2}$	14.275096
Median	Middle value	-	74.5
IQR	Interquartile Range	$Q3 - Q1$	22
Mode	Most frequent value	-	No mode

Additional info: These procedures and statistics are foundational for business analytics, enabling data-driven decision making and interpretation of business data sets.