Calculate measures of spread

Measures of spread quantify the variability of numerical data and complement measures of center. The learner calculates and interprets the range, interquartile range, variance, and standard deviation for ungrouped data, understanding that range and IQR describe positional spread, whereas variance and standard deviation summarize deviations from the mean; variance uses squared units, standard deviation uses the original units, and outliers can affect these measures differently. Inferential estimation and advanced distributional measures are beyond this scope.

Detailed Explanation: Calculate measures of spread

Measures of spread describe how much the data values vary. We will use the data set

2,\ 4,\ 6,\ 8,\ 10,\ 12

Assume these values are the complete data set, so we use the population variance formula.

1. Find the range

The range is the largest value minus the smallest value:

\text{Range}=12-2=10

The data spread across $10$ units.

2. Find the interquartile range

The data are already in order.

The lower half is $2,4,6$ , so $Q_1=4$ .
The upper half is $8,10,12$ , so $Q_3=10$ .

The interquartile range is

\text{IQR}=Q_3-Q_1=10-4=6

The middle $50\%$ of the data spreads across $6$ units.

3. Find the variance

First calculate the mean:

\bar{x}=\frac{2+4+6+8+10+12}{6}=\frac{42}{6}=7

Now find each deviation from the mean and square it:

\begin{array}{c|c|c} x & x-\bar{x} & (x-\bar{x})^2\\ \hline 2 & -5 & 25\\ 4 & -3 & 9\\ 6 & -1 & 1\\ 8 & 1 & 1\\ 10 & 3 & 9\\ 12 & 5 & 25 \end{array}

Add the squared deviations:

25+9+1+1+9+25=70

Divide by the number of values, $6$ :

\text{Variance}=\frac{70}{6}\approx 11.67

The variance is approximately $11.67$ square units.

4. Find the standard deviation

The standard deviation is the square root of the variance:

\text{Standard deviation}=\sqrt{11.67}\approx 3.42

The standard deviation is approximately $3.42$ units. This means the values are typically about $3.42$ units away from the mean of $7$ .

Interpreting the measures

Range: $10$ units; it uses only the smallest and largest values.
IQR: $6$ units; it describes the spread of the middle half of the data.
Variance: approximately $11.67$ square units; it uses squared deviations from the mean.
Standard deviation: approximately $3.42$ units; it describes spread in the original units.

The range can be strongly affected by an unusually small or large value. The IQR is less affected by outliers because it focuses on the middle $50\%$ . Variance and standard deviation are also affected by outliers because they measure distances from the mean, and squaring makes large deviations especially influential.

Learn by doing: Calculate measures of spread

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Statistics - Quartiles - Data Set (No Outliers) to Interquartile Range

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