Mean, Median, Mode

TB: 6.1

Mean, median and mode are ways to summarize a numerical dataset by calculating to a singular number.

Mean

The average of all numbers in the dataset.

The summation symbol

The mean formula:

mean is equal to the sum of all data points divided by 
the total number of data points.

$$mean = \frac{\sum_{}{} x_i}{n}$$

x_i is each data point at index i

n represents the total number of data points

Example

Poker earnings from 12 table games.
$12
$37
$5
$28
$44
$19
$7
$50
$22
$11
$33
$41

The following table above is the monetary earnings made from 12 poker games. We will be calculating the mean of this dataset.

Sum of all earnings (each data points all added up)

$$sum = 12 + 37 + 5 + 28 + 44 + 19 + 7 + 50 + 22 + 11 + 33 + 41 = 309$$

Total Number of data points: 12

$$mean = \frac{309}{12} = 25.75$$

Conclusion

The calculated mean equals 25.75.

The player made $309 in 12 table games, and on average the player made $25.75

Median

The middle-most data point in a dataset.

When calculating the median, the dataset must be sorted.

Steps to calculate the median:

1. Sort the dataset from least to greatest

2. If the data set has odd number of data points, the middle value is the median

3. If the data set has even number of data points, then calculate the mean of the two middle most values.

Example with odd number of data points

$$data = \{3, 1, 4, 1, 5\}$$

The following above contains 5 data points.

When sorted we get: {1, 1, 3, 4, 5}

The middle-most data point after the sorting is the value: 3.

Therefore, the median of the value is 3.

Example with even number of data points

$$poker = \{12,37,5,28,44 ,19,7,50,22,11,33,41\}$$

The following set above contains our previous 12 data points with our poker earnings. When sorted, we get following order:

$$poker = \{5, 7, 11, 12, 19, 22, 28, 33, 37, 41, 44, 50\}$$

In this dataset, there are even number of data points; therefore, we must calculate the mean of the two middle data points.

• 6th number: 22

• 7th number: 28

$$median = \frac{22+28}{2} = 25$$

Conclusion

In the data set of poker earnings, there are 6 data points total where the dollars earned was lower than the median of $25.

Since the data set has 12 data points, the remain 6 earnings were above the median.

Mode

The most occuring data point in a dataset.

A data set can have the following properties:

• Unimodal (one mode)

• Bimodal (two modes)

• Multimodal (more than two)

• Or have no mode at all.

Unimodal: Just one mode

$$data = \{1, 2, 2, 3, 4\}$$

The dataset above has a mode of 2.

2 was the most occurring data point, and it is unimodal no other data points were repeated.

Bimodal: two modes

$$data = \{3, 4, 4, 5, 6, 7, 7, 8\}$$

The dataset above has a mode of 4 and 7.

3 and 7 both occurred twice in the dataset. Hence, this dataset is bimodal.

Multimodal: more than two

$$data = \{2, 2, 3, 3, 4, 4, 5, 6\}$$

The dataset above has a mode of 2, 3, 4.

Values of 2, 3 and 4 occur twice in the dataset making the dataset multimodal.

No mode

$$data = \{1, 2, 3, 4, 5\}$$

The dataset has no repeating values; therefore, there is no mode.