Statistics

The One Thing

Statistics converts collected data into organised tables, visual representations, and summary measures so that patterns can be identified and reliable conclusions drawn. The representation and average selected must suit the type of data and the purpose of the investigation.

Definitions and Results

• Data: A collection of facts, numbers, or observations gathered for a particular purpose.
• Raw data: Data recorded in its original, unorganised form.
• Primary data: Information collected directly by the investigator through observation, measurement, surveys, or experiments.
• Secondary data: Information obtained from existing sources such as reports, records, or published tables.
• Qualitative data: Data describing qualities or categories, such as colour or type.
• Quantitative data: Numerical data that can be counted or measured.
• Discrete data: Data consisting of countable values, usually whole numbers, such as the number of students.
• Continuous data: Data that can take any value within a range, such as height, weight, or temperature.
• Frequency: The number of times a particular value or observation occurs.
• Frequency distribution: A table showing values or class intervals together with their frequencies.
• Class interval: A group or range used to organise continuous data, such as 10–20 or 20–30.
• Class size: The difference between the upper and lower boundaries of a class interval.
• Class mark: The midpoint of a class interval, calculated as

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

• Tally marks: Grouped strokes used to record and count frequencies quickly.
• Bar graph: A graph using separate rectangular bars to represent frequencies of distinct categories or values.
• Histogram: A graph using adjoining rectangles to represent frequencies of continuous class intervals.
• Frequency polygon: A line graph formed by joining points representing class marks and their frequencies.
• Pie chart: A circular diagram divided into sectors, where each sector represents a proportion of the total.
• Mean: The sum of all observations divided by the number of observations. For ungrouped data,

$$\text{Mean}=\frac{\text{sum of all observations}}{\text{number of observations}}.$$ For a frequency distribution, $$\text{Mean}=\frac{\Sigma fx}{\Sigma f},$$ where $x$ is the observation or class mark and $f$ is its frequency.

• Median: The middle value after data has been arranged in ascending or descending order. If $n$ is odd, the median is the $\left(\frac{n+1}{2}\right)$th observation. If $n$ is even, it is the average of the $\left(\frac{n}{2}\right)$th and $\left(\frac{n}{2}+1\right)$th observations.
• Mode: The observation that occurs most frequently in a data set.
• Range: The difference between the greatest and least observations:

$$\text{Range}=\text{greatest observation}-\text{least observation}.$$

• Outlier: A value unusually far from the other observations that may strongly affect the mean.
• Pie-chart angle: For a sector representing frequency $f$ out of total frequency,

$$\text{central angle}=\frac{\text{frequency}}{\text{total frequency}}\times 360^{\circ }.$$ The total of all sector angles is $360^{\circ }$.

• Graph conventions: Every statistical graph should have a suitable title, labelled axes, an appropriate scale, and clearly marked units. A bar graph has gaps between bars, whereas a histogram has no gaps because its intervals are continuous.
• Frequency total: The sum of all frequencies gives the total number of observations.
• Interpretation of averages: Averages are representative values and do not describe every individual observation. The mean uses every observation and is sensitive to extreme values; the median is less affected by extreme values; and the mode identifies the most common value.
• Reliability of conclusions: Data should be collected carefully and presented honestly. Representative samples and unbiased data collection improve the reliability of statistical results.

Worked Methods

1. Organising raw data into a frequency distribution

• Begin with the raw data in its original, unorganised form.
• Identify each distinct value or, for continuous data, suitable class intervals such as $10\text{–}20$ and $20\text{–}30$.
• Use tally marks to record each occurrence.
• Convert the tally marks into numerical frequencies.
• Check that the sum of all frequencies equals the total number of observations.
• For grouped continuous data, calculate the class mark of each interval using

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

Organising raw data into a frequency table reduces confusion and makes comparisons easier.

2. Constructing a bar graph

• Identify the distinct categories or values and their frequencies.
• Give the graph a suitable title.
• Label the axes and include the appropriate units.
• Select a scale that represents the frequencies accurately.
• Draw separate rectangular bars with heights corresponding to the frequencies.
• Leave gaps between the bars because the categories or values are distinct.

3. Constructing a histogram

• Organise the continuous data into class intervals and frequencies.
• Label the axes, provide a title, and select an appropriate scale.
• Represent each class interval by a rectangle whose height corresponds to its frequency.
• Place the rectangles next to one another without gaps because the intervals are continuous.

4. Constructing a frequency polygon

• Prepare a frequency distribution for the grouped data.
• Calculate the class mark for each class interval:

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

• Plot each point with coordinates $(\text{class mark},\text{frequency})$.
• Join consecutive points with straight line segments.
• Label the axes, units, scale, and graph title clearly.

5. Constructing a pie chart

• Find the total frequency.
• Calculate the angle for each category using

$$\text{central angle}=\frac{\text{frequency}}{\text{total frequency}}\times 360^{\circ }.$$

• Draw a circle.
• Use a protractor to mark each calculated sector angle.
• Label the sectors and provide a suitable title.
• Check that the sector angles total $360^{\circ }$.

6. Calculating the mean

For ungrouped data:

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

$$\text{Mean}=\frac{\text{sum of all observations}}{\text{number of observations}}.$$

For a frequency distribution:

• List each observation or class mark $x$.
• Record its frequency $f$.
• Calculate $fx$ for each row.
• Find $\Sigma fx$ and $\Sigma f$.
• Calculate

$$\text{Mean}=\frac{\Sigma fx}{\Sigma f}.$$

For grouped continuous data, use the class mark as the representative value of each class.

7. Calculating the median

• Arrange the observations in ascending or descending order.
• Count the number of observations, $n$.
• If $n$ is odd, select the $\left(\frac{n+1}{2}\right)$th observation.
• If $n$ is even, identify the $\left(\frac{n}{2}\right)$th and $\left(\frac{n}{2}+1\right)$th observations and calculate their average.

8. Identifying the mode and calculating the range

To find the mode:

• Determine the frequency of each observation.
• Identify the observation with the highest frequency.
• That observation is the mode.

To find the range:

• Identify the greatest observation.
• Identify the least observation.
• Subtract:

$$\text{Range}=\text{greatest observation}-\text{least observation}.$$

Where It Goes Wrong

• Raw data may be used without first being organised into a frequency table, making comparisons more difficult.
• Class marks may be omitted or calculated incorrectly when finding the mean for grouped continuous data.
• The median is often found without arranging the observations first; the odd- and even-$n$ conditions must be distinguished.
• A bar graph may be drawn without gaps, or a histogram with gaps. Bars are separated for distinct categories, whereas histogram intervals are continuous.
• Pie-chart angles may not be calculated from the total frequency, or the sector angles may fail to add to $360^{\circ }$.
• Graphs may lack a title, labelled axes, a suitable scale, or clearly marked units, which can produce misleading interpretations. Extreme values may also be overlooked even though they can strongly affect the mean.

What Gets Asked

• Define and distinguish data types, including primary and secondary, qualitative and quantitative, and discrete and continuous data.
• Convert raw data into a frequency distribution using tally marks and calculate the total frequency.
• Identify class intervals, class size, and class marks.
• Construct or interpret a bar graph, histogram, frequency polygon, or pie chart.
• Calculate pie-chart sector angles using

$$\frac{\text{frequency}}{\text{total frequency}}\times 360^{\circ }.$$

• Calculate the mean of ungrouped data and the mean of a frequency distribution using

$$\frac{\Sigma fx}{\Sigma f}.$$

• Find the median after arranging data and apply the correct formula for odd or even $n$.
• Identify the mode and calculate the range.
• Compare the mean, median, and mode and select the most suitable measure for a particular data set or purpose.
• Explain the effect of an outlier on the mean and why the median may be less affected.
• Interpret statistical graphs and assess whether their titles, labels, scales, units, and spacing are appropriate.
• Explain why careful, unbiased data collection and representative samples are necessary for reliable statistical conclusions.