Location
Location identifications refer to key figures describing the "average size" or quality of the values in the data. For numerical data, they can thought as location at number line.
Mode (mo) is the most typical value of a variable. It is not necessarily unique but can be used with all scales. In Data 1, there are two boys with the height of 147 cm and two boys with the height of 149 cm. There are two modes in this case and they are 147 cm and 149 cm.
Median (md) is the middlemost value of an ordered data. If number of observations is even, then median is the average of two middlemost values. Median cannot be determined for variables in nominal scale. In Data, median is 150 cm as it is thirteenth in the ordered data presented in Table 2.
Mean is the most known statistical measure. It is very sensitive for extreme values. For that reason, it should not be informed alone. For example, one university informed that mean salary of their newly graduated students were very high in one program compared to overall mean salary of that field. They did not mention, that one of those graduated was NBA player earning millions. This one graduated upgraded mean to be high, although median was low meaning that 50% of graduated had low salary. This NBA player is called extreme value or outlier. All these must be checked in the data and it also must be decided, whether they are included or excluded. If they are included, trimmed mean or winsorized mean would be better. In trimmed mean, some percentage of data in both ends is excluded. In winsorized mean, some percentage of data is replaced with the first (last) value the of bottom (top) end included in the data. In the trimmed mean, the number of observations is decreasing, whereas in winsorized mean it remains the same. In both bases, outliers are not affecting as much in the normal mean.
For discrete values, mean is solved with the formula
In Data 1:
The mean height is this case would be 152 cm.
The 5 % trimmed mean means that 5% from both ends is excluded. As 5% from 25 is 1.25, it means one observation is excluded form both ends (138 and 172).
The winsorized mean would be
In this case, all of them are very close to each other meaning, that there is no outliers.
If some values are more important than the others, then values could be weighted. This kind of case could be, for example, solving mean of grades. Each course give some credits and grades are multiplied with them. Divisor in this case would be the sum of credits. The weighted mean is solved as follows:
In the following table, there is credits and grades of three courses for one student.
Weighted mean is
In Figure 1 is presented mode, median and mean with Normal distribution. In the ideal situation, they are almost the same and graph follows Normal distribution like in case a. If mode and median are less than mean, then the data is focused on the first part of the distribution (case b). This means, that there are outliers at the end. If they are more than mean (case c), the data is focused on the end and the outliers are at the first part. If mean and median are about the same and there are two modes (case d), it indicates that there may be two different normal-distributed groups in the same variable. It should not be used in analysis.
Quartiles
Lower quartile (q1) and upper quartile (q3) are location indicators at 25% and 75% of the ordered data. Actually, median is quartile at 50%. As 25% of the Data 1 is 6.25, the seventh value of the ordered data is lower quartile At the same way, the nineteenth value of the ordered data is the upper quartile