Arithmetic Mean

Sharing a Total Equally

Besides the median (middle value) and mode (most frequent value), the mean, or average, is another way to describe the center of a data set.

The mean is the value each member would receive if we distributed the total evenly. Imagine collecting all your friends' candies into one pile and then sharing them equally. The number each friend receives is the mean.

How to calculate the mean

To calculate the mean:

1. Add all the data values.
2. Divide the sum by the number of values.

Mean formula:

The quantities in this formula are:

• $\bar{x}$, read "x bar," represents the mean.
• $\sum x$, read "sigma x," represents the sum of all data values.
• $n$ is the number of data values.

Effect of Large Values on Mean Median and Mode

We will compare the mean, median, and mode of the same data set before and after adding two unusually large values. The three measures do not react in the same way, and the comparison shows which one keeps describing the typical value.

Collecting Clothes for a Community Drive

Initial situation:

Ten members of School A's student council collect wearable clothes for a community drive. The number of items collected by each member is:

Finding the initial mean, median, and mode:

1. Sort the data (for the median and mode):

   $3, 3, 4, 5, 5, 6, 7, 8, 9, 10$ (there are $10 \text{ data points}$, $n = 10$)

2. Calculate the mean:

   $$\begin{aligned} \sum x &= 3+3+4+5+5 \\ &\quad+6+7+8+9+10 \\ &= 60 \end{aligned}$$

   $$\bar{x} = \frac{\sum x}{n} = \frac{60}{10} = 6$$

   So, the initial mean is $6$.

3. Find the median:

   The number of data points is even ($n = 10$). The middle data are at positions $10/2 = 5$ and $(10/2)+1 = 6$.

   The value at position $5$ is $5$, and the value at position $6$ is $6$.

   $$\text{Median} = \frac{5+6}{2} = 5.5$$

   So, the initial median is $5.5$.

4. Find the mode:

   The most frequent values are $3$ ($2 \text{ times}$) and $5$ ($2 \text{ times}$).

   So, the initial modes are $3$ and $5$ (bimodal).

New situation:

The next day, $2 \text{ students}$ joined and donated $20$ and $22$ items of clothing.

The new data becomes:

Finding the new mean, median, and mode:

1. Sort the data:

   $3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 20, 22$ (there are $12 \text{ data points}$, $n = 12$)

2. Calculate the new mean:

   $$\sum x = 60 + 20 + 22 = 102$$

   $$\bar{x} = \frac{\sum x}{n} = \frac{102}{12} = 8.5$$

   So, the new mean is $8.5$.

3. Find the new median:

   The number of data points is even ($n=12$). The middle data are at positions $12/2 = 6$ and $(12/2)+1 = 7$.

   The value at position $6$ is $6$, the value at position $7$ is $7$.

   $$\text{Median} = \frac{6+7}{2} = 6.5$$

   So, the new median is $6.5$.

4. Find the new mode:

   The most frequent values are still $3$ ($2 \text{ times}$) and $5$ ($2 \text{ times}$).

   So, the new modes are $3$ and $5$.

Impact of Adding Data

Compare the measures of central tendency before and after the two values are added:

• Mean: Changed from $6$ to $8.5$ (increased by $2.5$).
• Median: Changed from $5.5$ to $6.5$ (increased by $1$).
• Mode: Remained $3$ and $5$ (unchanged).

Adding $20$ and $22$, two values far above the original data, has the largest effect on the mean. The larger values pull the mean upward.

The median also changes, but less sharply. The mode does not change at all.

Impact of an Extreme Value

Replace the $12$th student's donation with $100 \text{ items}$, then recalculate the mean to measure the effect of this unusually large value.

Data becomes: $3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 20, 100$ ($n=12$)

1. Calculate the mean:

   $$\sum x = 60 + 20 + 100 = 180$$

   $$\bar{x} = \frac{\sum x}{n} = \frac{180}{12} = 15$$

   The mean becomes $15$, far above both the initial mean ($6$) and the previous mean ($8.5$).

2. Find the median:

   Sorted data: $3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 20, 100$

   The median is still $(6+7)/2 = 6.5$, the same as before.

3. Find the mode:

   Mode remains $3$ and $5$.

An extreme value may be classified as an outlier under a stated statistical rule or through domain judgment. In this example, $100$ strongly affects the mean but leaves the median and modes unchanged. This sensitivity is the mean's main weakness. The median and mode are more stable when one unusually large or small value appears.

The mean is easy to calculate, but an unusually large or small value can make it less representative of the rest of the data.