Averages and Measures of Spread : Mean

The mean is the sum of all data values divided by the number of values. It is also known as the arithmetic average.

Formula:

\( \text{Mean} = \dfrac{\text{Sum of values}}{\text{Number of values}} \)

The mean is best used when the data does not have extreme values (outliers) which can distort the average.

Note: The mean is affected by extreme values. If there are unusually large or small numbers in the data, consider using the median instead.

Averages and Measures of Spread : Median

The median is the middle value when data is arranged in numerical order. It is not affected by extreme values (outliers) and is useful for skewed data.

How to find the median:

• If the number of values is odd, the median is the middle number.
• If the number of values is even, the median is the average of the two middle numbers.

Median from a frequency table:

1. Find the cumulative frequency.
2. Locate the middle value ( \( \dfrac{n+1}{2} \) if individual data or \( \dfrac{n}{2} \) if grouped).
3. Identify the class that contains the median.

Note: The median is not affected by extreme values. It gives a better picture of typical values in skewed distributions.

Averages and Measures of Spread : Mode

The mode is the value that appears most frequently in a data set. A data set can have:

• One mode (unimodal)
• Two modes (bimodal)
• No mode (if all values occur equally)

Mode is especially useful for categorical or discrete data.

Note: The mode may not exist or may not be useful for continuous data unless data is grouped into intervals. It is best for identifying the most common value.

Averages and Measures of Spread : Quartiles

Quartiles divide a data set into four equal parts. They are useful for describing the spread and identifying outliers.

• Lower Quartile (Q1): The median of the lower half of the data (below the overall median).
• Upper Quartile (Q3): The median of the upper half of the data (above the overall median).
• Interquartile Range (IQR): The difference between Q3 and Q1: \( \text{IQR} = Q_3 – Q_1 \)

Quartiles are used to measure spread and detect skewness or outliers.

Tip: Use quartiles to find the IQR (spread of middle 50% of the data), especially useful when the data contains outliers.

Averages and Measures of Spread : Range and Interquartile Range

Measures of spread tell us how spread out the values in a data set are. Two common measures are:

• Range: Difference between the largest and smallest value
• Interquartile Range (IQR): Difference between the upper quartile (Q3) and the lower quartile (Q1):
  \( \text{IQR} = Q_3 – Q_1 \)

Use: IQR is a better measure of spread than range when data contains extreme values or outliers.

Tip: The IQR is not affected by outliers, so it is useful for comparing the spread of skewed data.