Statistics

The One Thing

Statistics turns collected data into meaningful information by organising, representing and interpreting it. Accurate analysis depends on matching the representation and measure of central tendency to the type of data and the purpose of the study.

Definitions and Results

• Data: A collection of facts, observations or numerical information gathered for a specific purpose.
• Raw data: Data collected in its original, unorganised form.
• Observation: Each individual value or item in a data set.
• Frequency: The number of times a particular observation or class occurs.
• Frequency distribution: A table showing observations or class intervals together with their frequencies.
• Ungrouped data: Data listed as individual observations without class intervals.
• Grouped data: Data arranged into class intervals, such as 0–10, 10–20 and 20–30.
• Class interval: A range of values used to group data.
• Class limits: The smallest and largest values included in a class interval.
• Class size: The difference between the upper and lower limits or boundaries of a class interval.
• Class mark: The midpoint of a class interval:

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

• Bar graph: A graph using separate rectangular bars to represent frequencies of discrete categories. Gaps are usually left between bars.
• Histogram: A graph of grouped continuous data using adjoining rectangles. Their widths represent class intervals and their heights represent frequencies.
• Frequency polygon: A graph formed by joining points representing class marks and corresponding frequencies with straight line segments.
• Arithmetic mean: The sum of all observations divided by the number of observations.
• Median: The middle value after observations have been arranged in ascending or descending order.
• Mode: The observation occurring most frequently in a data set.
• Central tendency: A single value representing the centre or typical value of a data set.
• Inclusive class interval: A class interval that includes both its lower and upper limits, such as 10–19.
• Exclusive class interval: A class interval in which the upper limit is excluded from that class and included in the next, such as 10–20.

• Mean of ungrouped data: For observations $x_1,x_2,\dots ,x_n$,

$$\text{mean}=\frac{x_1+x_2+\cdots +x_n}{n}.$$

• Mean of a frequency distribution:

$$\text{mean}=\frac{\sum f_i{x}_i}{\sum f_i},$$ where $x_i$ is an observation and $f_i$ is its frequency. The total frequency $\sum f_i$ is the total number of observations.

• Mean of grouped data: Use the class mark $x_i$ as the representative value of each class interval in

$$\text{mean}=\frac{\sum f_i{x}_i}{\sum f_i}.$$

• Median of $n$ ungrouped observations: Arrange the observations in 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: Identify the value with the greatest frequency. A data set may have one mode, more than one mode or no mode.
• Bar graphs and histograms: Bar-graph bars are separated because categories are distinct. Histogram rectangles touch because class intervals represent continuous data.
• Axes and scale: The horizontal axis generally shows observations, categories or class intervals; the vertical axis shows frequency. A suitable scale must be chosen and stated clearly.
• Class boundaries: For continuous class intervals with gaps, such as 10–19 and 20–29, subtract $0.5$ from each lower limit and add $0.5$ to each upper limit. The intervals become 9.5–19.5 and 19.5–29.5.
• Frequency polygon: Plot class marks against frequencies and join the plotted points.
• Histogram area: When class widths are equal, the total area of the rectangles is related to the total frequency.
• Effect of transformations on the mean: If every observation is increased or decreased by the same number, the mean changes by that number. If every observation is multiplied or divided by a fixed non-zero number, the mean changes in the same ratio.
• Interpretation: Conclusions should address comparisons, trends, highest and lowest frequencies, and what the central value suggests about the data.

Worked Methods

Organising data in a frequency distribution

• Collect the raw data in its original, unorganised form.
• Identify the individual observations or suitable class intervals.
• Count how many times each observation or class occurs.
• Record these counts as frequencies.
• For grouped data, state the class intervals, class limits, class size and, where required, the class marks.
• Check that the total frequency $\sum f_i$ equals the total number of observations.

Constructing a bar graph

• Identify the discrete categories or observations.
• Place the categories on the horizontal axis.
• Place frequency on the vertical axis.
• Choose and clearly state a suitable scale.
• Draw separate rectangular bars with heights equal to the corresponding frequencies.
• Leave gaps between the bars because the categories are distinct.

Constructing a histogram

• Arrange the continuous data into class intervals, such as 0–10, 10–20 and 20–30.
• If intervals have gaps, convert them to class boundaries. For 10–19 and 20–29, use 9.5–19.5 and 19.5–29.5.
• Place class intervals or boundaries on the horizontal axis.
• Place frequency on the vertical axis.
• Choose and state a suitable scale.
• Draw adjoining rectangles whose widths represent the class intervals and whose heights represent frequencies.
• Ensure that the rectangles touch, since the data are continuous.

Constructing a frequency polygon

• Identify each class interval.
• Calculate its class mark using

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

• Pair each class mark with its corresponding frequency.
• Plot the class marks on the horizontal axis and the frequencies on the vertical axis.
• Join the plotted points with straight line segments.
• Use the resulting polygon to represent or compare distributions.

Calculating the mean of ungrouped data

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

$$\text{mean}=\frac{x_1+x_2+\cdots +x_n}{n}.$$

Calculating the mean from a frequency distribution

• List each observation $x_i$ and its frequency $f_i$.
• Calculate $f_i{x}_i$ for each observation.
• Find $\sum f_i{x}_i$.
• Find $\sum f_i$, the total frequency.
• Calculate

$$\text{mean}=\frac{\sum f_i{x}_i}{\sum f_i}.$$

Calculating the mean of grouped data

• List the grouped class intervals and their frequencies.
• Calculate the class mark $x_i$ for every interval.
• Multiply each class mark by its frequency to obtain $f_i{x}_i$.
• Calculate $\sum f_i{x}_i$ and $\sum f_i$.
• Use

$$\text{mean}=\frac{\sum f_i{x}_i}{\sum f_i}.$$ The class mark is treated as the representative value of each class.

Calculating the median

• Arrange ungrouped 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.
• Average these two observations to obtain the median.

Finding the mode

• Count the frequency of each observation or category.
• Identify the value with the greatest frequency.
• Report that value as the mode.
• State whether the data have one mode, more than one mode or no mode.

Interpreting a table or graph

• Check the title, labels, class intervals, frequencies and scale.
• Identify the highest and lowest frequencies.
• Compare categories or distributions.
• Describe relevant trends.
• Consider whether unusually large or small values affect the interpretation.
• Explain what the central value suggests about the data.
• Interpret the summary in context rather than treating it as a complete description of every observation.

Where It Goes Wrong

• Treating a histogram like a bar graph: histogram rectangles must touch for continuous grouped data, whereas bar-graph bars are separated for distinct categories.
• Forgetting to arrange observations before finding the median, or using the odd-number rule when the number of observations is even.
• Using class limits directly instead of class marks when calculating the mean of grouped data.
• Omitting the total frequency $\sum f_i$ from the denominator of

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

• Ignoring class boundaries for intervals such as 10–19 and 20–29; the corresponding continuous boundaries are 9.5–19.5 and 19.5–29.5.
• Interpreting a graph without checking its scale, labels, frequencies or extreme values.

What Gets Asked

• Define data, raw data, observation, frequency, frequency distribution, grouped data and ungrouped data.
• Distinguish between class intervals, class limits, class size and class mark.
• Construct or interpret a frequency distribution.
• Draw or identify a bar graph for discrete categories.
• Draw or identify a histogram for grouped continuous data, including the use of class boundaries.
• Construct a frequency polygon by plotting class marks against frequencies.
• Calculate the mean of ungrouped data using

$$\frac{x_1+x_2+\cdots +x_n}{n}.$$

• Calculate the mean from a frequency distribution or grouped table using

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

• Find the median for odd and even numbers of observations.
• Identify the mode and determine whether a data set has one mode, more than one mode or no mode.
• Explain how the mean, median and mode respond to extreme values and when each is useful.
• Determine how the mean changes when every observation is increased, decreased, multiplied or divided by a fixed number.
• Interpret tables and graphs by comparing values, identifying trends and highest or lowest frequencies, and explaining the significance of the central value.