Statistics

The One Thing

Statistics describes the centre and distribution of data using the mean, median and mode. Histograms display frequencies across continuous class intervals, while ogives display cumulative frequencies and allow estimates of the median and quartiles.

Definitions and Results

• Data: A collection of numerical or descriptive information gathered for study.
• Raw data: Data recorded in its original, unarranged form.
• Frequency: The number of times a particular value or class occurs.
• Class interval: A range of values used to group data, such as 10–20 or 20–30.
• Class width: The difference between the upper and lower boundaries of a class interval.
• Class mark: The midpoint of a class interval:

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

• Mean: The average obtained by dividing the sum of all observations by the number of observations.
• Mean of ungrouped data: For observations $x_1,x_2,\dots ,x_n$,

$$\bar{x}=\frac{\Sigma x}{n}.$$

• Mean of a frequency distribution: For values $x_i$ with frequencies $f_i$,

$$\bar{x}=\frac{\Sigma (f_i{x}_i)}{\Sigma f_i}.$$

• Mean of grouped data: For grouped class intervals, use the class marks $x_i$:

$$\bar{x}=\frac{\Sigma (f_i{x}_i)}{\Sigma f_i}.$$

• Assumed mean method: If $A$ is an assumed mean and $d_i=x_i-A$,

$$\bar{x}=A+\frac{\Sigma (f_i{d}_i)}{\Sigma f_i}.$$

• Step-deviation method: If $h$ is the common class width and

$$u_i=\frac{x_i-A}{h},$$ then $$\bar{x}=A+h\left[\frac{\Sigma (f_i{u}_i)}{\Sigma f_i}\right].$$

• Median: The middle value when observations are arranged in ascending or descending order.
• Median of grouped data: For a continuous frequency distribution,

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

• Mode: The value that occurs most frequently in a data set.
• Modal class: The class interval having the greatest frequency.
• Mode of grouped data:

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

• Empirical relation: For a moderately symmetrical distribution,

$$\text{Mode}\approx 3\text{Median}-2\text{Mean}.$$ This is an approximation, not an exact rule for every distribution.

• Histogram: A graph for continuous grouped data formed from adjoining rectangles, with class intervals on the horizontal axis and frequencies on the vertical axis.
• Frequency density: For unequal class widths,

$$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$$ Rectangle height is based on frequency density so that rectangle area represents frequency.

• Class boundaries: Adjusted boundaries used to convert inclusive intervals into continuous intervals. For whole-number data, this commonly involves subtracting $0.5$ from lower limits and adding $0.5$ to upper limits.
• Cumulative frequency: The running total of frequencies up to a particular class.
• Less-than cumulative frequency: The total frequency below each upper class boundary; it is used to draw a less-than ogive.
• More-than cumulative frequency: The total frequency above each lower class boundary; it is used to draw a more-than ogive.
• Ogive: A cumulative-frequency curve plotted using class boundaries and cumulative frequencies.
• Median from ogives: Locate $N/2$ on the cumulative-frequency axis, draw a horizontal line to the ogive, and drop a perpendicular to the data axis. The corresponding data-axis value estimates the median.
• Quartiles: Values dividing ordered data into four equal parts. $Q_1$ corresponds to $N/4$, and $Q_3$ corresponds to $3N/4$, on a cumulative-frequency graph.
• Interquartile range: The spread of the middle half of the data:

$$\text{Interquartile range}=Q_3-Q_1.$$

The mean uses every observation, the median depends on position, and the mode depends on frequency. A symmetric distribution generally has $$\text{Mean}=\text{Median}=\text{Mode}.$$

Worked Methods

1. Finding the mean of ungrouped data

• Add all observations.
• Count the number of observations, $n$.
• Divide the sum by $n$:

$$\bar{x}=\frac{\Sigma x}{n}.$$

2. Finding the mean of frequency data

• List each value $x_i$ and its frequency $f_i$.
• Calculate $f_i{x}_i$ for each value.
• Find $\Sigma (f_i{x}_i)$ and $\Sigma f_i$.
• Apply

$$\bar{x}=\frac{\Sigma (f_i{x}_i)}{\Sigma f_i}.$$ The total frequency is $$N=\Sigma f.$$

3. Finding the mean of grouped data

• Identify the class intervals and their frequencies.
• Calculate each class mark:

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

• Multiply each class mark by its frequency.
• Use

$$\bar{x}=\frac{\Sigma (f_i{x}_i)}{\Sigma f_i}.$$

4. Using the assumed mean method

• Choose an assumed mean $A$, usually a convenient class mark.
• Calculate

$$d_i=x_i-A.$$

• Calculate $f_i{d}_i$ for each class.
• Apply

$$\bar{x}=A+\frac{\Sigma (f_i{d}_i)}{\Sigma f_i}.$$

5. Using the step-deviation method

• Choose an assumed mean $A$.
• Use the common class width $h$.
• Calculate

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

• Calculate $f_i{u}_i$.
• Apply

$$\bar{x}=A+h\left[\frac{\Sigma (f_i{u}_i)}{\Sigma f_i}\right].$$

6. Finding the median of raw data

• Arrange the observations in ascending or descending order.
• If $n$ is odd, select the observation at position

$$\frac{n+1}{2}.$$

• If $n$ is even, average the observations at positions

$$\frac{n}{2}\text{and}\frac{n}{2}+1.$$

7. Finding the median of a discrete frequency distribution

• Calculate the cumulative frequencies.
• Determine the median position from the total frequency.
• Locate the value corresponding to that position in the cumulative-frequency table.

8. Finding the median of grouped data

• Calculate the total frequency $N=\Sigma f$.
• Calculate $N/2$.
• Use cumulative frequencies to identify the median class: it is the class whose cumulative frequency first reaches or exceeds $N/2$.
• Identify:

- $l$, the lower 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

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

9. Finding the mode

• For ungrouped data, identify the value occurring most frequently.
• For grouped data, identify the modal class as the class interval with the highest frequency.
• If an estimated numerical mode is required, identify $l,f_1,f_0,f_2$ and $h$, then use

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

10. Constructing a histogram

• Convert inclusive class intervals into continuous class boundaries where necessary. For whole-number data, subtract $0.5$ from lower limits and add $0.5$ to upper limits.
• Place class intervals on the horizontal axis and frequency on the vertical axis.
• Draw adjoining rectangles with no gaps because the data are continuous.
• If class widths are equal, use frequencies directly as rectangle heights.
• If class widths are unequal, calculate

$$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}$$ and use frequency density for the heights.

• Use suitable scales, label both axes, include units, and plot accurately.
• If required, form a frequency polygon by joining the midpoints of the tops of the histogram rectangles.

11. Constructing a less-than ogive

• Calculate less-than cumulative frequencies.
• Plot each upper class boundary against its less-than cumulative frequency.
• Join the plotted points smoothly.
• Label both axes and use an appropriate scale.

12. Constructing a more-than ogive

• Calculate more-than cumulative frequencies.
• Plot each lower class boundary against its more-than cumulative frequency.
• Join the plotted points smoothly.
• Label both axes and use an appropriate scale.

13. Estimating the median and quartiles from ogives

• For a single ogive, locate $N/2$ on the cumulative-frequency axis.
• Draw a horizontal line to the ogive.
• Drop a perpendicular to the data axis.
• Read the corresponding value as the estimated median.
• To estimate $Q_1$, use $N/4$ on the cumulative-frequency axis.
• To estimate $Q_3$, use $3N/4$.
• If both less-than and more-than ogives are drawn on the same axes, their point of intersection gives an approximate median; the corresponding horizontal-axis value is the median.
• Calculate the interquartile range using

$$Q_3-Q_1.$$

Where It Goes Wrong

• The median is found before arranging raw observations; the data must first be placed in ascending or descending order.
• For an even number of observations, only one middle observation is selected instead of averaging the observations at positions $n/2$ and $(n/2)+1$.
• In grouped-data calculations, class marks, frequencies, cumulative frequencies and the total frequency $N=\Sigma f$ are not identified correctly.
• The grouped median formula is used without selecting the correct median class or without interpreting $l$, $cf$, $f$, and $h$ correctly.
• Histograms are drawn with gaps, or unequal class widths are represented by frequency rather than frequency density; continuous data require adjoining rectangles, and unequal widths require

$$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}.$$

• Ogives are plotted against class limits instead of the appropriate class boundaries, or upper and lower boundaries are reversed for less-than and more-than cumulative frequencies.

What Gets Asked

• Calculate the mean of ungrouped data using

$$\bar{x}=\frac{\Sigma x}{n}.$$

• Calculate the mean of a frequency distribution or grouped data using

$$\bar{x}=\frac{\Sigma (f_i{x}_i)}{\Sigma f_i}.$$

• Find a grouped mean using the assumed mean method or the step-deviation method.
• Find the median of raw data, including separate cases for odd and even numbers of observations.
• Calculate the median of a discrete frequency distribution from cumulative frequencies.
• Calculate the median of grouped data using

$$l+\left[\frac{N/2-cf}{f}\right]\times h.$$

• Identify the mode or modal class and calculate the estimated grouped-data mode.
• Use the empirical relation

$$\text{Mode}\approx 3\text{Median}-2\text{Mean}.$$

• Construct a histogram from grouped continuous data, including cases with unequal class widths and frequency density.
• Convert inclusive class intervals into continuous class boundaries.
• Construct a frequency polygon from histogram rectangle midpoints.
• Calculate less-than and more-than cumulative frequencies.
• Draw less-than and more-than ogives using upper and lower class boundaries respectively.
• Estimate the median from a single ogive or from the intersection of two ogives.
• Estimate $Q_1$, $Q_3$, and the interquartile range $Q_3-Q_1$ from a cumulative-frequency graph.
• Interpret which measure of central tendency is most appropriate: the mean for overall numerical calculations, the median when position or extreme values matter, and the mode when the most common value is required.