Statistics

The One Thing

For grouped data, calculate the mean from class marks and frequencies, locate the median through cumulative frequency, and determine the mode from the modal class and its neighboring frequencies. Correct use of class boundaries is essential for the median and mode formulas.

Definitions and Results

• Grouped data: Data arranged in class intervals together with the frequency of observations in each interval.
• Class interval: A range of values, such as 10–20 or 20–30, used to organize data.
• Lower and upper class limits: The smallest and largest stated values of a class interval.
• Class boundaries: Values that make adjoining class intervals continuous. For example, 10–19 becomes 9.5–19.5.
• Class width: The difference between the upper and lower class boundaries, usually denoted by $h$.
• Class mark: The midpoint of a class interval:

$$x_i=\frac{\text{lower class limit}+\text{upper class limit}}{2}.$$

• Frequency: The number of observations in a particular class interval, denoted by $f_i$.
• Cumulative frequency: The running total of frequencies. It is used to locate the median class.
• Mean: A measure of central tendency found by dividing the total of all observations by their number.
• Median: The value that divides an ordered data set into two equal parts.
• Mode: The value or class that occurs most frequently.
• Median class: The class interval whose cumulative frequency is greater than or equal to $N/2$, where

$$N=\sum f_i.$$

• Modal class: The class interval with the highest frequency.
• Direct mean:

$$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}.$$

• Assumed mean method:

$$\bar{x}=a+\frac{\sum f_i{d}_i}{\sum f_i},d_i=x_i-a,$$ where $a$ is the assumed mean.

• Step-deviation method:

$$\bar{x}=a+h\left(\frac{\sum f_i{u}_i}{\sum f_i}\right),u_i=\frac{x_i-a}{h}.$$

• Grouped-data median:

$$\text{Median}=l+\left[\frac{(N/2)-cf}{f}\right]h,$$ where $l$ is the lower class boundary of the median class, $cf$ is the cumulative frequency before the median class, $f$ is the frequency of the median class, and $h$ is the class width.

• Grouped-data mode:

$$\text{Mode}=l+\left[\frac{f_1-f_0}{2f_1-f_0-f_2}\right]h,$$ where $l$ is the lower class boundary of the modal class, $f_1$ is its frequency, $f_0$ is the frequency of the preceding class, $f_2$ is the frequency of the succeeding class, and $h$ is the class width.

• Empirical relationship: For a moderately skewed distribution,

$$\text{Mode}=3(\text{Median})-2(\text{Mean}).$$

• Interpretation: The mean uses every observation and may not be an actual observation. The median and mode are generally less affected by extreme values; the median is suitable for skewed data, while the mode identifies the most common value or category.

Worked Methods

Finding the class mark

• Identify the lower and upper class limits.
• Add them.
• Divide by 2:

$$x_i=\frac{\text{lower limit}+\text{upper limit}}{2}.$$

• For the class interval 10–20, the class mark is

$$x_i=\frac{10+20}{2}=15.$$

Calculating the mean by the direct method

• For each class, calculate the class mark $x_i$.
• Multiply each class mark by its frequency to obtain $f_i{x}_i$.
• Calculate the total frequency:

$$N=\sum f_i.$$

• Calculate the total of the products:

$$\sum f_i{x}_i.$$

• Apply

$$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}.$$

Calculating the mean by the assumed mean method

• Select an assumed mean $a$, usually a convenient class mark.
• Calculate the deviation for each class:

$$d_i=x_i-a.$$

• Multiply each deviation by its frequency to obtain $f_i{d}_i$.
• Add the products and frequencies.
• Apply

$$\bar{x}=a+\frac{\sum f_i{d}_i}{\sum f_i}.$$

Calculating the mean by the step-deviation method

• Select an assumed mean $a$.
• Determine the class width $h$.
• Calculate

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

• Multiply each coded deviation by its frequency to obtain $f_i{u}_i$.
• Apply

$$\bar{x}=a+h\left(\frac{\sum f_i{u}_i}{\sum f_i}\right).$$

Finding the median of grouped data

• If the class intervals are inclusive, such as 10–19 and 20–29, convert them into continuous intervals. Thus, 10–19 becomes 9.5–19.5.
• Calculate the total frequency:

$$N=\sum f_i.$$

• Calculate $N/2$.
• Construct the cumulative frequencies.
• Identify the median class: the class whose cumulative frequency is greater than or equal to $N/2$.
• Identify:

- $l$, the lower class boundary of the median class; - $cf$, the cumulative frequency before the median class; - $f$, the frequency of the median class; - $h$, the class width.

• Substitute into

$$\text{Median}=l+\left[\frac{(N/2)-cf}{f}\right]h.$$

The median depends on the order and cumulative frequencies of the data, not merely on the most frequent class.

Finding the mode of grouped data

• If the class intervals are inclusive, convert them into continuous intervals before using the formula.
• Identify the modal class, which has the highest frequency.
• Identify:

- $l$, the lower class boundary of the modal class; - $f_1$, the frequency of the modal class; - $f_0$, the frequency of the preceding class; - $f_2$, the frequency of the succeeding class; - $h$, the class width.

• Substitute into

$$\text{Mode}=l+\left[\frac{f_1-f_0}{2f_1-f_0-f_2}\right]h.$$

The mode is determined by the class with the greatest frequency, but its formula also requires the frequencies of the neighboring classes.

Using the empirical relationship

For a moderately skewed distribution, if two of the three measures are known, calculate the third using $$\text{Mode}=3(\text{Median})-2(\text{Mean}).$$

Selecting an appropriate measure

• Use the mean when overall numerical balance is required and reliable frequency information is available.
• Use the median for skewed data or data containing extreme values.
• Use the mode when the most common value or category is required.

Where It Goes Wrong

• Class marks are sometimes omitted or calculated incorrectly; the mean formulas require

$$x_i=\frac{\text{lower limit}+\text{upper limit}}{2}.$$

• The total frequency $N=\sum f_i$ must be calculated before locating $N/2$ for the median.
• The median class is identified using cumulative frequency, not simply by choosing the class with the highest frequency.
• Inclusive intervals such as 10–19 and 20–29 must be converted into continuous class boundaries before applying the median or mode formula.
• The median formula requires the cumulative frequency before the median class, $cf$, rather than the cumulative frequency of the median class.
• The mode formula requires $f_0$, $f_1$, and $f_2$: the frequencies of the preceding, modal, and succeeding classes.

What Gets Asked

• Define grouped data, class intervals, class limits, class boundaries, class width, class marks, frequency, and cumulative frequency.
• Calculate class marks for given class intervals.
• Find the grouped-data mean using the direct method.
• Find the grouped-data mean using the assumed mean method.
• Find the grouped-data mean using the step-deviation method.
• Convert inclusive class intervals such as 10–19 and 20–29 into continuous intervals.
• Construct cumulative frequencies and identify the median class.
• Calculate the median using

$$\text{Median}=l+\left[\frac{(N/2)-cf}{f}\right]h.$$

• Identify the modal class and calculate the mode using

$$\text{Mode}=l+\left[\frac{f_1-f_0}{2f_1-f_0-f_2}\right]h.$$

• Use

$$\text{Mode}=3(\text{Median})-2(\text{Mean})$$ for a moderately skewed distribution.

• Explain why the mean, median, or mode is the most suitable measure for a particular data set.