Averages from Tables | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Averages from Tables & Charts 

How do we find averages if there are lots of values?

• In practical scenarios, the volume of data available for analysis is typically extensive, extending beyond a few isolated figures.
• To facilitate comprehension, this data is commonly structured in formats like tables or charts, enhancing readability and interpretation.
• When presented in a tabular or graphical form, it remains possible to calculate statistical measures such as the mean, median, and mode.
• However, a crucial aspect involves grasping the underlying message conveyed by the table or chart.

How do I find averages from a table or chart?

• Finding the median and mode from tables/charts is quite simple once you grasp the information presented in the table or chart. 
  • Tables are essential for summarizing data neatly and, importantly, they arrange the data in order.

Finding the mean from (discrete) data presented in tables:

• Finding the mean from discrete data presented in tables involves understanding the information conveyed by the table. 
• Tables provide the data value (e.g., the number of pets per household) and the frequency of that data value (e.g., the number of households with that number of pets).
• The mean can be determined once you understand the information in the table. Tables provide data values and their frequencies.

Steps to Calculate Mean from Table Data:

• Step 1: Add a column to the table to calculate 'data value' multiplied by 'frequency'. This step involves the preliminary computation required for determining the mean.
• Step 2: Calculate the total of the additional column to get the sum of all data values in the table.
• Step 3: Find the mean by dividing the total data value by the sum of frequencies, essentially dividing the overall data values by the count of data points.

Finding the median from (discrete) data presented in tables:

• Examine the data table to pinpoint the median's position accurately.
• The position of the median can be found by using where n is the number of data values
• Use the table to deduce where the value lies
  • For instance, if the median corresponds to the 7th value and the cumulative frequencies of the preceding rows are 4 and 7, the median will be within the 7 values of the subsequent row.

Finding the mode (or modal value)

Identifying the mode (or modal value) is a straightforward process.

• Look for the value with the highest frequency in the data set.
  • This high frequency corresponds to a specific data value.
• Ensure not to mistake the data value for the frequency itself.
  • In a table, the frequency guides us to the row containing the mode.

Averages from Grouped Data

• When dealing with certain scenarios, data can exhibit significant variation. For instance, consider the heights of individuals, especially when the dataset comprises both children and adults.
• Data like height is considered continuous, meaning it can be measured. However, listing every single height value, especially in a mixed dataset, can be cumbersome.
• Due to minimal differences between adjacent values (e.g., someone being 176 cm tall versus 177 cm tall), it is common practice to group such data into classes.
• Grouping data into classes is a practical approach, but it comes with a trade-off: the loss of raw data. For example, knowing that 10 individuals have heights between 150 cm and 160 cm does not reveal the exact heights of those 10 people.
• Consequently, when data is grouped, the original raw data is obscured. This makes it challenging to calculate the precise mean, median, and mode using their traditional definitions. However, estimations can be made, especially when the term "estimate" is mentioned in questions.
  • We can estimate the mean and identify the class interval within which the median falls.
  • Similarly, we can determine the modal class, which denotes the class interval containing the mode.

How do I find/estimate the mean from grouped data?

• There is one extra stage to this method compared to finding the mean from tables with discrete data.
• We use the class midpoints as our data values. For example, if heights are split into class intervals 150 ≤ x < 160, 160 ≤ x < 170, etc., the midpoints, and so data values, would be 155, 165, etc.
  • Step 1:
    • Draw an extra two columns on the end of a table of the grouped data.
    • In the first new column, write down the midpoint of each class interval.
  • Step 2:
    • Work out "frequency" × "midpoint" (This is often called fx).
  • Step 3:
    • Total the fx column, and if not already done nor mentioned in the question, total the frequency column to find the number of data values involved.
  • Step 4:
    • Estimate the mean by using its formula: "total of fx" ÷ "no. of data values".
• Be careful with midpoints
• Not all class intervals will be of equal size, so there may not be a nice pattern to the midpoints.

How do I find the class interval that the median lies in?

• Find the median position by calculating n/2, where 𝑛n is the total number of data values (sum of the frequency column).
• Use the frequency table to locate the class interval that contains the (n/2)^th value.
  • For example, if the median position is the 7th value and the frequencies of the first two class intervals are 4 and 7 respectively, the median lies in the second class interval.
• When discussing the median in the context of grouped data, refer to the "class interval containing the median" rather than simply "the median."

How do I find the modal class interval?

• Focus on the class interval that contains the modal value, which is the class interval with the highest frequency.
• Locate the highest frequency in the table.
• Identify the class interval associated with this highest frequency.
• Ensure not to confuse the class interval with the frequency itself; the frequency indicates which class interval contains the mode.