To find the median of ungrouped data, the values or measurements must be arranged in an array. It may be either in an increasing or a decreasing oreder. Then we get the value or the measurement in the middle. If there is an even number of values, then the two values in the middle are added together then divided by two.

Example 1:
Data / Measurements
78 95 93 88 75 99 71 72 74 85 87 88 95 86 87 84 82

Array of the Data / Measurements
71 72 74 75 78 82 84 85 86 87 87 88 88 93 95 95 99

The middle value is 86. The median is 86.

Example 2:
Data / Measurements
12 18 14 17 9 11 20 21 15 13

Array of the Data / Measurements
9 11 12 13 14 15 17 18 20 21

The middle values are 14 and 15. The median is (14+15)/2 = 14.5.

An array is an arrangement of values in increasing or decreasing order.

e.g.
Data / Measurements
78 95 93 88 75 99 71 72 74 85 87 88 95 86 87 84 82

Array of the Data / Measurements
71 72 74 75 78 82 84 85 86 87 87 88 88 93 95 95 99

Below are the characteristics of the mean of any distribution:
1. The mean is the most appropriate measure of central tendency when the data are in the interval or ratio scale.
2. The mean lies between the largest and smallest values or measurements.
3. There is only one value for the mean for a given set of values or measurements.
4. The mean is easily influenced by extreme values because all values contribute to the average. If there are high values, the mean tends to be high also. If there are extremely low values, the mean tends to be low also.

where:
X_am = assumed mean
f = frequency
d = coded deviation
n = total frequency
i = class size

To find the mean using the coded formula, the following procedure may be employed:
1. Choose any class interval as the assumed mean. The class mark of this class interval will be our
2. Construct the column for the deviation (d). This is done by putting 0 as the deviation for our assumed mean (deviation from itself). For the class intervals larger than the class interval which contains the assumed mean, the deviations are 1, 2, 3, and so on. For the class intervals smaller than the class interval which contains the assumed mean, the deviations are -1, -2, -3, and so on in that order.
3. Multiply each frequency by the corresponding deviation to get the entries in the fd column.
4. Get the sum of the value obtained in step 3 to obtain
5. Use the formula to compute the mean.

The formula for the mean using the classmark is as follows:

where:
x = measurement or score
f = frequency
X = classmark
n = total frequency

The steps in computing the mean using the midpoint formula are as follows:
1. Construct the column for the classmark (X). The class mark is the midpoint of the lower and upper limits of a given class interval. The midpoint is founded by getting one-half of the sum of the upper limit and the lower limit.
2. Multiply each classmark by its corresponding frequency. This will be entered in the column for fX.
3. Get the sum of the values obtained in Step 2 to get ∑fx.
4. Substitute the obtained values in Step 3 in the formula to find the mean.

There are sets of values where the values or measurements involved have different “weights” or degree of importance. The mean of this set is called a “weighted mean.” The formula for computing a weighted mean is as follows:

where:
= mean
x = measurement or value
W = weight

Example:
Ina’s subjects and the corresponding number of units and grades.

To compute for the weighted mean, multiply each value (in this case the grade) with the corresponding weight (in the example, the number of units). Get the sum of all the products then divide it by the total number of weights (sum of all units).

Example:
The Sales of Nena’s Sari-Sari Store for one week.

To get the mean (or average) sales for the week, we add all the sales figures for each day (x), and divide it by the number of days (n).

Although text, tables and graphs effectively present data, these do not sufficiently describe the data being presented.

Measures of Central Tendency - numerical descriptive measures which indicate the center of a data set

There are three commonly used measures of central tendency: