Changing the mean

Introduction to Changing the Mean

• The mean is the average value in a set of data. It’s calculated by adding all values and dividing by the number of values.
• A powerful characteristic of the mean is its sensitivity to changes in the distribution’s values. This feature allows statisticians to explore ‘what-if’ scenarios by simulating changes in the data.
• Two types of change are common: adding a constant to all values or multiplying all values by a constant. Both have distinct, predictable effects on the mean.

Adding a Constant to All Values

• When a constant is added (or subtracted) from each value in the data set, the mean changes by the same constant.
• This is because adding a constant to all values changes the total sum used to calculate the mean by (constant) x (number of entries), which when divided by the number of entries gives the constant.
• Example: if you add 3 to all values, the mean also increases by 3.

Multiplying All Values by a Constant

• When each value in the data set is multiplied (or divided) by a constant, the mean also multiplies (or divides) by the same constant.
• This is because multiplying all values scales the total sum calculated for the mean, and when divided by the number of entries results in the same scaling factor.
• Example: if you multiply all values by 3, the mean also increases 3 times.

Potential Misconceptions About Changing the Mean

• It’s crucial to understand that although adding or multiplying constants to all values changes the mean, this doesn’t mean the shape of the data distribution changes. The distribution simply shifts location or scale.
• Changing the mean doesn’t affect the shape of the distribution, which means changes to the mean do not affect its variances or standard deviation.
• The variance and standard deviation provide information about the spread of the data, which remains the same when adding a constant, but increases or decreases when multiplying by a constant.

Further Points on Changing the Mean

• The behaviours of mean under addition and multiplication are supportive foundations for many further statistical concepts.
• These principles are not just for theoretical exploration but are also practical in diverse fields such as economics, psychology, and engineering for predicting outcomes under proposed changes.
• An in-depth understanding of these principles is essential for succeeding in statistical units of Further Maths syllabus, and especially in the Further Stats 1 section.