**Averages
**An average is a value representing a group of data. The common averages are the mean, median and mode.

**Mean
**Mean is the most common, we know the average, comparing the summary of values. It is the sum of all the numbers divided by the number of addends or respondents.

*Example:*

The scores of ten students are 82, 85, 79, 78, 89, 87, 88, 89, 75, and 77. What is the mean?

\( Mean = \frac{sum\ of\ all\ numbers}{number\ of\ respondents}\)

\( Mean = \frac{82 + 85 + 79 + 78 + 89 + 87 + 88 + 89 + 75 + 77}{10}\)

\( Mean = \frac{829}{10}\)

Mean = 82.9 are 83

82.9 or 83 is the average. It can be in decimal or as a whole number.

**Median
**The median is the value of the middle term when data are arranged. It can be in ascending or descending order. In computing the median, remember the following:

a. Arrange the number in numerical order.

b. If the items are an odd number, the median is in the middle.

c. If the items are an even number, compute the central number.

**Median with the odd number
**

*Example:*

The number of books students borrowed each day are 63, 31, 42, 54 and 50.

a. Arrange the following 31, 42, 50, 54, 63.

b. The number of items is odd because there are 5 items. The median is 50.

**Median with an even number
**

*Example:*

The scores of ten students are 82, 85, 79, 78, 89, 87, 88, 89, 75, and 77. What is the median?

a. Arrange the following 75, 77, 78, 79, 82, 85, 87, 88, 89, 89

b. There are 10 students. The middle terms are 82 and 85. To compute for the median

\( Median = \frac{82 + 85}{2} = 33.5\)

**Mode
**The mode is referred to as the most frequent numbers that occur in the set of data. It requires no calculation to determine the value.

*Example:*

The scores of ten students are 82, 85, 79, 78, 89, 87, 88, 89, 75, and 77. What is the mode?

The mode is 89.

The \(\underline{range}\) is described to be the difference between the highest and lowest numbers. To calculate the range, find the lowest and highest numbers, then subtract.

*Example:
*89 – 75 = 14

Therefore, the range is 14.

**Grouped Data
**Calculate the average income for 40 days on threads of The Threads and Needles Shop.

The above examples the data is grouped. It is a long method of calculations because of the complex calculations that may result in large values. Adding up the numbers of sales divided by the number of days may not result in finding the mean. Since, in the first row, the data tells that 3 days they have a sale of 172 -180, but the exact sales of each day is not provided.

a.To estimate the mean, compute for the midpoint of each amount of sales.

\( x = \frac{172 -180}{2} = 176 \)

\( x = \frac{163 -171}{2} = 167 \)

\( x = \frac{154 -162}{2} = 158\)

\( x = \frac{145 -153}{2} = 149\)

\( x = \frac{136 -144}{2} = 140\)

\( x = \frac{127 -135}{2} = 131\)

\( x = \frac{118 -126}{2} = 122\)

b. Then multiply the number of days (f) and the midpoint (x) of each sales.

We can know estimate using the sum of fx divide by number of days

\( mean = \frac{∑fx}{Σf}\)

\( mean = \frac{6041}{40}\)

mean = 151.03

The mean is 151.03