Statistics

The One Thing

Measures of dispersion quantify how widely observations are spread around a central value. Range provides a simple estimate, whereas mean deviation, variance, and standard deviation give progressively more detailed measures; the coefficient of variation enables relative comparison between data sets.

Definitions and Results

• Dispersion: The extent to which data values differ from one another or spread around a central value.
• Range: The difference between the largest and smallest observations in a data set. For ungrouped data,

$$\text{Range}=\text{Largest observation}-\text{Smallest observation}.$$ For a frequency distribution using class limits, $$\text{Range}=\text{Upper limit of the highest class}-\text{Lower limit of the lowest class}.$$

• Coefficient of Range: A relative measure of dispersion:

$$\frac{L-S}{L+S},$$ where $L$ is the largest value and $S$ is the smallest value.

• Deviation: The difference between an observation and a chosen central value:

$$x-A.$$

• Mean Deviation: The arithmetic mean of the absolute deviations from a central value such as the mean, median, or mode.
• Absolute Deviation: The non-negative deviation

$$|x-A|,$$ because positive and negative deviations are treated as distances.

• Mean Deviation for ungrouped data:

$$MD(A)=\frac{\sum |x_i-A|}{n}.$$

• Mean Deviation for a frequency distribution:

$$MD(A)=\frac{\sum f_i|x_i-A|}{\sum f_i}.$$ Mean deviation may be calculated about the mean, median, or mode; among these, mean deviation about the median is generally the least.

• Variance: The arithmetic mean of the squares of deviations from the arithmetic mean.
• Variance for ungrouped data:

$${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}.$$

• Variance for a frequency distribution:

$${\sigma }^2=\frac{\sum f_i(x_i-\bar{x}{)}^2}{\sum f_i}.$$

• Standard Deviation: The positive square root of variance:

$$\sigma =\sqrt{{\sigma }^2}.$$ For ungrouped data, $$\sigma =\sqrt{\frac{\sum (x_i-\bar{x}{)}^2}{n}}.$$ For a frequency distribution, $$\sigma =\sqrt{\frac{\sum f_i(x_i-\bar{x}{)}^2}{\sum f_i}}.$$

• Coefficient of Variation: A relative measure of dispersion:

$$\frac{\text{standard deviation}}{\text{mean}}\times 100.$$ If the means are positive, the data set with the smaller coefficient of variation is considered more consistent.

• Discrete Frequency Distribution: A distribution in which each distinct value and its frequency are given.
• Continuous Frequency Distribution: A distribution in which observations are grouped into class intervals with corresponding frequencies. For grouped continuous data, $x_i$ represents the class mark or midpoint of the class interval.
• Sum of deviations from the arithmetic mean:

$$\sum (x_i-\bar{x})=0.$$

• Units: Variance is expressed in squared units, whereas standard deviation is expressed in the same units as the observations.
• Non-negativity: Standard deviation is always non-negative and is zero only when all observations are equal.
• Transformation properties: Adding or subtracting the same constant from every observation does not change variance or standard deviation. Multiplying every observation by a constant $k$ multiplies standard deviation by $|k|$ and variance by $k^2$.

Worked Methods

1. Calculating the range

• Identify the largest observation.
• Identify the smallest observation.
• Subtract the smallest observation from the largest:

$$\text{Range}=\text{Largest observation}-\text{Smallest observation}.$$

For a frequency distribution using class limits:

• Identify the upper limit of the highest class.
• Identify the lower limit of the lowest class.
• Subtract the latter from the former:

$$\text{Range}=\text{Upper limit of highest class}-\text{Lower limit of lowest class}.$$

2. Calculating the coefficient of range

• Let $L$ be the largest value and $S$ the smallest value.
• Calculate $L-S$.
• Calculate $L+S$.
• Divide:

$$\text{Coefficient of Range}=\frac{L-S}{L+S}.$$

3. Calculating mean deviation for ungrouped data

To calculate mean deviation about a chosen value $A$:

• Select $A$, which may be the mean, median, or mode.
• Calculate each deviation $x_i-A$.
• Convert each deviation to an absolute deviation $|x_i-A|$.
• Add the absolute deviations.
• Divide by the number of observations:

$$MD(A)=\frac{\sum |x_i-A|}{n}.$$

The absolute values are required because positive and negative deviations represent distances and must not cancel.

4. Calculating mean deviation for a frequency distribution

• Select the central value $A$.
• For each value or class mark $x_i$, calculate $|x_i-A|$.
• Multiply each absolute deviation by its frequency $f_i$.
• Add the products.
• Divide by the total frequency:

$$MD(A)=\frac{\sum f_i|x_i-A|}{\sum f_i}.$$

For grouped continuous data, use the class mark or midpoint as $x_i$.

5. Calculating variance and standard deviation directly

For ungrouped data:

• Calculate the arithmetic mean $\bar{x}$.
• Calculate each deviation $x_i-\bar{x}$.
• Square each deviation:

$$(x_i-\bar{x}{)}^2.$$

• Add the squared deviations.
• Divide by $n$:

$${\sigma }^2=\frac{\sum (x_i-\bar{x}{)}^2}{n}.$$

• Take the positive square root:

$$\sigma =\sqrt{{\sigma }^2}.$$

For a frequency distribution:

• Calculate the mean $\bar{x}$.
• Calculate each deviation $x_i-\bar{x}$.
• Square each deviation.
• Multiply by the corresponding frequency $f_i$.
• Add the weighted squared deviations.
• Divide by the total frequency:

$${\sigma }^2=\frac{\sum f_i(x_i-\bar{x}{)}^2}{\sum f_i}.$$

• Take the positive square root:

$$\sigma =\sqrt{\frac{\sum f_i(x_i-\bar{x}{)}^2}{\sum f_i}}.$$

Squaring prevents positive and negative deviations from cancelling and gives greater weight to larger deviations.

6. Using the shortcut formula

For ungrouped data:

• Calculate $\sum x_i^2$.
• Divide by $n$.
• Calculate the mean $\bar{x}$ and square it.
• Subtract:

$${\sigma }^2=\frac{\sum x_i^2}{n}-{\bar{x}}^2.$$

For a frequency distribution:

• Calculate $\sum f_i{x}_i^2$.
• Divide by $\sum f_i$.
• Calculate ${\bar{x}}^2$.
• Subtract:

$${\sigma }^2=\frac{\sum f_i{x}_i^2}{\sum f_i}-{\bar{x}}^2.$$

Then calculate $\sigma =\sqrt{{\sigma }^2}$.

7. Using the assumed-mean method

If $$d_i=x_i-A,$$ where $A$ is an assumed mean:

• Choose a convenient assumed mean $A$.
• Calculate $d_i=x_i-A$.
• Calculate $f_i{d}_i$ and $f_i{d}_i^2$.
• Substitute into:

$${\sigma }^2=\frac{\sum f_i{d}_i^2}{\sum f_i}-{\left(\frac{\sum f_i{d}_i}{\sum f_i}\right)}^2.$$

• Take the positive square root to obtain $\sigma$.

8. Using the step-deviation method

If $$u_i=\frac{x_i-A}{h},$$ where $A$ is an assumed mean and $h$ is the common class width or scaling factor:

• Choose an assumed mean $A$.
• Calculate the class mark $x_i$, where necessary.
• Calculate

$$u_i=\frac{x_i-A}{h}.$$

• Calculate $f_i{u}_i$ and $f_i{u}_i^2$.
• Substitute into:

$${\sigma }^2=h^2\left[\frac{\sum f_i{u}_i^2}{\sum f_i}-{\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)}^2\right].$$

• Take the positive square root to obtain standard deviation.

9. Comparing consistency

• Calculate the standard deviation for each data set.
• If the means or units differ, calculate the coefficient of variation:

$$CV=\frac{\sigma }{\bar{x}}\times 100.$$

• Compare the coefficients of variation.
• Provided the means are positive, the smaller coefficient of variation indicates greater consistency.

A smaller standard deviation generally indicates that observations are more concentrated around their mean, but standard deviation alone may be misleading when means or units differ.

Where It Goes Wrong

• Using only the largest and smallest observations for a measure intended to use all observations: range is strongly affected by extreme values and does not describe the full distribution.
• Forgetting absolute values when calculating mean deviation: $|x_i-A|$ is required so that positive and negative deviations do not cancel.
• Dividing a frequency calculation by $n$ instead of the total frequency $\sum f_i$.
• Using the observations directly for grouped continuous data instead of using the class mark or midpoint as $x_i$.
• Forgetting to square deviations when calculating variance, even though squaring prevents cancellation and gives greater weight to larger deviations.
• Comparing standard deviations alone when the data sets have different means or units; the coefficient of variation provides the required relative comparison.

What Gets Asked

• Define dispersion, range, coefficient of range, deviation, absolute deviation, mean deviation, variance, standard deviation, coefficient of variation, discrete frequency distribution, and continuous frequency distribution.
• Calculate the range and coefficient of range for ungrouped observations or a frequency distribution.
• Calculate mean deviation about the mean, median, or mode for ungrouped data.
• Calculate mean deviation for a frequency distribution, including grouped continuous data using class marks.
• Calculate variance and standard deviation directly for ungrouped data.
• Calculate variance and standard deviation for a frequency distribution.
• Apply the shortcut formulas:

$${\sigma }^2=\frac{\sum x_i^2}{n}-{\bar{x}}^2$$ and $${\sigma }^2=\frac{\sum f_i{x}_i^2}{\sum f_i}-{\bar{x}}^2.$$

• Use the assumed-mean method:

$${\sigma }^2=\frac{\sum f_i{d}_i^2}{\sum f_i}-{\left(\frac{\sum f_i{d}_i}{\sum f_i}\right)}^2.$$

• Use the step-deviation method:

$${\sigma }^2=h^2\left[\frac{\sum f_i{u}_i^2}{\sum f_i}-{\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)}^2\right].$$

• Explain why the sum of deviations from the arithmetic mean is zero:

$$\sum (x_i-\bar{x})=0.$$

• Explain why deviations are squared in variance.
• State the units and non-negativity properties of variance and standard deviation.
• Determine the effect of adding or subtracting a constant, or multiplying observations by $k$, on variance and standard deviation.
• Compare the consistency of two data sets using standard deviation or, where means or units differ, the coefficient of variation.