Statistics

The One Thing

Statistics summarises numerical data through measures of central tendency and position, especially the median, quartiles, and mode. The correct method depends on whether the data are ungrouped, discrete frequency data, or continuous grouped data, and requires accurate ordering, cumulative frequencies, and class boundaries.

Definitions and Results

• Statistics: The collection, arrangement, presentation, analysis, and interpretation of numerical data.
• Raw Data: Data collected in its original, unarranged form.
• Array: Data arranged in ascending or descending order.
• Median: The middle value of an ordered data set. For $n$ observations:

- if $n$ is odd, $\text{Median}$ is the value of the $\frac{n+1}{2}$th observation; - if $n$ is even, $\text{Median}=\frac{1}{2}\left[\text{value of the }\frac{n}{2}\text{th observation}+\text{value of the }\left(\frac{n}{2}+1\right)\text{th observation}\right]$.

• Median of a Discrete Frequency Distribution: After calculating cumulative frequencies, the median is the value corresponding to the $\frac{N+1}{2}$th observation, or commonly the $\frac{N}{2}$th observation, where $N$ is the total frequency.
• Median of a Continuous Frequency Distribution:

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

• Quartiles: Three values dividing ordered data into four equal parts: $Q_1$, $Q_2$, and $Q_3$.
• First Quartile ($Q_1$): The value below which approximately 25% of observations lie.
• Second Quartile ($Q_2$): The median; approximately 50% of observations lie below it.
• Third Quartile ($Q_3$): The value below which approximately 75% of observations lie.
• Quartile Deviation:

$$QD=\frac{Q_3-Q_1}{2}$$

• Interquartile Range: The range of the middle 50% of observations:

$$IQR=Q_3-Q_1$$

• Quartile Coefficient of Dispersion:

$$\frac{Q_3-Q_1}{Q_3+Q_1}$$ provided $Q_3+Q_1\ne 0$.

• Mode: The observation or value occurring most frequently.
• Mode of a Discrete Distribution: The value with the greatest frequency.
• Modal Class: The class interval with the highest frequency.
• Mode of a Continuous Distribution:

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

• Cumulative Frequency: The running total of frequencies, obtained by adding each frequency to all preceding frequencies.
• Less-than Cumulative Frequency: Cumulative frequency formed from the smallest class upward.
• More-than Cumulative Frequency: Cumulative frequency formed from the largest class downward.
• Empirical Relation: For a moderately skewed distribution,

$$\text{Mode}\approx 3\text{Median}-2\text{Mean}$$ This is approximate and is most reliable for moderately skewed, unimodal distributions.

• Skewness: The degree to which a distribution is asymmetrical:

- positively skewed: generally $\text{Mean}>\text{Median}>\text{Mode}$; - negatively skewed: generally $\text{Mode}>\text{Median}>\text{Mean}$.

• Open-ended Distribution: A distribution in which one or both extreme class intervals lack clearly stated lower or upper limits.
• Inclusive and Exclusive Class Intervals: Inclusive intervals contain both stated limits. Exclusive intervals use the upper limit as the boundary of the next class. Inclusive intervals may require conversion to continuous class boundaries before grouped-data formulas are applied.
• Frequency Density: For unequal class intervals, frequency density is

$$\text{Frequency density}=\frac{\text{frequency}}{\text{class width}}.$$ Frequencies should not be compared directly when class widths are unequal.

• Interpretive Results: The median is less affected by extreme values than the arithmetic mean and is therefore suitable for skewed data and open-ended distributions. The mode is useful for identifying the most common size, category, or choice and may be used with qualitative data. A data set may have one mode, more than one mode, or no mode.

Worked Methods

1. Median of ungrouped data

• Arrange the observations in an array, in ascending or descending order.
• Count the observations to obtain $n$.
• If $n$ is odd, select the value of the $\frac{n+1}{2}$th observation.
• If $n$ is even, average the values of the $\frac{n}{2}$th and $\left(\frac{n}{2}+1\right)$th observations.
• Report the result as the median.

2. Median of a discrete frequency distribution

• Construct the frequency table.
• Calculate the total frequency:

$$N=\sum f.$$

• Calculate the cumulative frequencies.
• Locate the $\frac{N+1}{2}$th observation, or commonly the $\frac{N}{2}$th observation.
• Identify the data value corresponding to that cumulative-frequency position.
• Interpret this value as the median.

3. Median of a continuous frequency distribution

• Calculate the total frequency $N$.
• Determine the median position, $N/2$.
• Construct the cumulative-frequency column.
• Identify the median class: the class whose cumulative frequency first reaches or exceeds $N/2$.
• Establish:

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

• Substitute into:

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

• Use class boundaries consistently, particularly for inclusive intervals.

The median may also be located graphically from a less-than ogive.

4. Quartiles of continuous grouped data

• Calculate the total frequency $N$.
• Locate the quartile positions:

$$Q_1:\frac{N}{4},Q_2:\frac{N}{2},Q_3:\frac{3N}{4}.$$

• Use cumulative frequencies to identify the relevant quartile class for each position.
• For each quartile, identify $l$, $c$, $f$, and $h$.
• Apply:

$$Q_1=l+\left[\frac{N/4-c}{f}\right]h,$$ $$Q_2=l+\left[\frac{N/2-c}{f}\right]h,$$ $$Q_3=l+\left[\frac{3N/4-c}{f}\right]h.$$

• Calculate the spread if required:

$$QD=\frac{Q_3-Q_1}{2},IQR=Q_3-Q_1.$$

• Calculate the quartile coefficient of dispersion when required:

$$\frac{Q_3-Q_1}{Q_3+Q_1},$$ provided $Q_3+Q_1\ne 0$.

Quartiles may also be obtained from the corresponding cumulative-frequency graph.

5. Mode of a discrete distribution

• List the values and their frequencies.
• Identify the greatest frequency.
• Select the value associated with that frequency.
• If several values share the greatest frequency, report more than one mode. If no value occurs more frequently than the others, the distribution has no mode.

6. Mode of a continuous distribution

• Identify the modal class, which has the highest frequency.
• Establish:

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

• If class intervals are unequal, compare frequency densities rather than raw frequencies to identify the modal class.
• Substitute into:

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

• Interpret the result as an estimated mode.

7. Empirical relation and skewness

• Use the approximate relation:

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

• Apply it only when the distribution is moderately skewed and unimodal.
• Identify the direction of skewness using:

- positive skewness: $\text{Mean}>\text{Median}>\text{Mode}$; - negative skewness: $\text{Mode}>\text{Median}>\text{Mean}$.

• Interpret the central tendency together with measures of dispersion, since a central value alone does not show how widely the observations are spread.

Where It Goes Wrong

• Failing to arrange raw data before finding the median; the median is determined by position in an ordered array.
• Using the wrong median position for an even or odd number of observations.
• Omitting the calculation of $N=\sum f$ or using ordinary frequencies instead of cumulative frequencies for discrete distributions.
• Using class limits instead of consistent lower class boundaries in continuous grouped-data formulas, especially for inclusive class intervals.
• Forgetting that $Q_1$, $Q_2$, and $Q_3$ correspond approximately to $N/4$, $N/2$, and $3N/4$, respectively.
• Comparing frequencies directly when class intervals are unequal, rather than using frequency density to identify the modal class.

What Gets Asked

• Define raw data, an array, median, quartiles, mode, cumulative frequency, modal class, skewness, and open-ended distribution.
• Find the median of an ordered ungrouped data set with odd or even $n$.
• Find the median of a discrete frequency distribution using cumulative frequencies.
• Calculate the median of a continuous frequency distribution using

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

• Calculate $Q_1$, $Q_2$, and $Q_3$ for continuous grouped data.
• Calculate quartile deviation, interquartile range, and quartile coefficient of dispersion.
• Identify the mode of a discrete distribution.
• Identify the modal class and calculate the mode of a continuous distribution.
• Determine the modal class when class intervals are unequal using frequency density.
• Construct or interpret less-than and more-than cumulative frequencies.
• Locate the median from a less-than ogive and quartiles from a cumulative-frequency graph.
• Use the empirical relation $\text{Mode}\approx 3\text{Median}-2\text{Mean}$ for a moderately skewed, unimodal distribution.
• Interpret whether a distribution is positively or negatively skewed.
• Explain why the median may be preferred for skewed or open-ended data and why the mode is useful for qualitative data.
• Explain why measures of central tendency should be interpreted alongside measures of dispersion.