Question

Intro. to Business Intelligence

List and briefly define the central tendency measures of descriptive statistics.

(well answered please)

Answer 1

To represent a dataset, we cannot use all the values of the dataset all the time. If we use all the values of the dataset representation purpose, it becomes cumbersome for large datasets. Even in cases, when we want to compare the datasets, it becomes hard because the datasets can be of different sizes.
In these cases, the central measure becomes important to represent the entire dataset by a single value. This single value is nothing but a summary of the entire dataset. The central value tells about a value that is ideally located at the center of the various data points in the dataset. This central value is also helpful in identifying the outliers in a dataset by comparing the data points in the dataset with this central value. The values with an extreme separation from this data point can be called as outlies. Similarly, these are also used to compare the different datasets. There are multiple mathematical tests that are based on these central values. The tests need not compare every data point in the datasets with each other, instead, we can compare these central values.

The three most commonly used central tendency measures are:

• Mean
• Median
• Mode

Mean:  
Mean is one of the most commonly used measures. It is also known as average. It can be calculated by dividing the summation of all the numbers in the dataset by the total count of numbers in the dataset.
For e.g. a dataset with numbers such as 32, 32, 32, 33 ,33, 34, 37,39,51
The sum of the digits is 323 and the count of digits in the dataset is 9.
The mean will be nothing but the sum of the digits in the dataset divided by the count of digits in the dataset that will be
323 divided by 9 or 323 / 9 i.e.35.89

The representation of mean for the population is given by

Here the summation of x represents the sum of all the numbers in the dataset while N represents the count of the numbers in the dataset.

For the sample, the mean is represented by M

The mean representation is susceptible to problems if there are outliers in the data. The ideal value of the mean gets skewed either at the high or low side based on the outlier values and their frequency. Generally, the mean is not one of the values of the dataset. It is best for continuous data.

Median:

The median gives the exact center value of the data. In the case of the dataset, the median value lies at the center of the dataset. There are equal values, above the median and equal values below the median. For the purpose of calculating the median, the dataset needs to be arranged from low to high values.
For e.g. our earlier dataset with numbers such as 32, 32, 32, 33 ,33, 34, 37,39,51
This dataset is already sorted. The dataset has 9 digits so there is an absolute central value. We will just use the 5th value i.e. 33 as the median. As we can see there are 4 values that lie on the left of the dataset while there are 4 values that lie on the right of the dataset. In case, if the dataset would have had 10 values, we would have to calculate the mean of the 5th and 6th value to get the median value.
Median works well for the datasets with outliers.

Mode:

The mode is nothing but a measure that gives the value with the highest frequency as the mode value.
For e.g. our earlier dataset with numbers such as 32, 32, 32, 33 ,33, 34, 37,39,51
Here 32 has frequency value as 3 while 33 has a frequency value 2. All other digits have a frequency value of 1. As we can see, 32 has the highest frequency among all these numbers. Thus the mode value is 32.

This measure is especially good in discrete data. To depict the mode in the continuous measure, we will have to club the data in different buckets such as 0-10, 10-20, etc. For all these buckets, we can have frequency and see, which bucket has the highest frequency.
The problem with the mode is in cases where the frequency measures of multiple numbers are the same. In those cases, it is hard to decide upon the true mode value of the dataset.