• Measures of Central Tendency in Statistics

## Measures of Central Tendency in Statistics

If you want to get some vital information from your data at a glance, the measures of central tendency are really a place to look. The median, mode, and range are the most widely used measures of central tendency in Statistics. They can also be perceived as different forms of averages or midpoints in a sample or data set. They are appropriate for different kinds of situations. They give better insight into the structure of a population or sample and possible connections between the data points. For instance, a range would easily show you the gap between the highest and the lowest points in your data or a sample.

### Understanding Measures of Central Tendency

Mean

The mean is simply the average. It gives you a point estimate into the variable you’re measuring. Let’s say you have information on the age of students in a class. The mean will quickly give you an idea of how old a student in that class would be. It is simply computed by taking the sum of all data points in your sample or population and dividing it by the sample size or population size, respectively. It is usually denoted by (pronounced x-bar) where x is the individual observation. Note: A sample is a sub-selection of the population; this selection could be either random or systematic e.g. The ages of 10 students in a certain class are given as follows; 13, 14, 18, 12, 15,20,13,17,19,16 To find the mean, first, we add up the ten numbers 13+14+18+12+15+20+13+17+19+16 = 157 and then divide by the total number of observations, which is 10 in this case, and we have Mean=157/10=15.7 Hence the average age of students in that class is 15.7 years, so could say that the class is a class of teenagers.

Life application:

All measures of central tendency are very important in solving and understanding real-life situations. For instance, a household makes use of different electrical, which consumes power at different rates. The appliances and their power usage are given below: Ceiling fan – 100 watts Laptop – 50 watts Fridge – 1000 watts TV – 100 watts Calculate the average power consumption of the household. Step 1: add the total power consumed; 100+50+1000+100 = 1250 watts Step 2: Divide the total power consumed by the number of appliances Number of appliances is 4, hence the mean=1250/4 =312.5 watts Therefore, the average power consumption of the household is 312.5 watts.

Median

This is another important measure of central tendency; the median is the number at the center or middle of a set of numbers that have been in ascending or descending order of magnitude. The position/value of this median is affected by the number of observations in the data set. When the number of observations is odd, the median would simply be the number at the middle, but if the number of observations is even, then two numbers would be in the middle, simply calculate the mean of both numbers. To do that, simply add them and divide by 2. e.g.,Find the median in the above set of numbers used for the calculation. The first step is to arrange all the observations into ascending order of magnitude; 12,13,13,14,15,16,17,18,19,20 Since the number of observations 10 is even, we take the average of the two words highlighted above, so we obtain (15+16)/2=15.5 If the last value from the observations leaving us with 12,13,13,14,15,16,17,18,19 the median would simply be 15. Note: There is a difference in the mean and median for our example, but it is also possible they are the same. In any case, the closer these measures, the better.

Life application

In the example cited above on power consumption, the median power consumption would be 100 watts. Note that the 100 watts are the average of the 2 100watts because upon the arrangement of the power consumption in ascending order of magnitude, we have 50w,100w,100w,1000w. so that the average of both numbers is still 100 watts

Mode

The mode is another interesting measure of central tendency. The mode in a set of observations is the observation or number that occurred the most in the data. It is also pertinent to note that two different observations can be a mode. In that case, we say that the observation is bimodal. In the same vein, an observation with 3 modes is said to be trimodal. Continuing with our example above, the mode of the observations 13, 14, 18, 12, 15,20,13,17,19,16 would be 13 since 13 appeared twice as highlighted while all the other numbers appeared only once. If the observations were tweaked to become 13, 14, 18, 12, 15,20,13,17,19,17, the observations would be bimodal with 13 and 17 being the mode.

Life Application:

The mode can help with the identification of baselines. From the data presented above on power consumption, the modal power consumption is 100 watts, which implies that most of the household appliances consume 100 watts.

Range

The range of a set of observations is the difference between the highest and the lowest value in the observation. Using our previous data once again, 13, 14, 18, 12, 15,20,13,17,19,16, the range is obtained as follows 20 – 12 = 8 where 20 is the highest observation, while 12 is the smallest value.

Life Application:

The range is a very useful measure of central in real-life applications. The range can be used to identify outliers. Using the appliance power consumption introduced above, the range for power consumption would be 1000 watts – 50=950. The range shows the power consumption of appliances varies. A value below or above a range in an electrical system can set off a warning alarm. It shows the validity of the other measures. For the example considered, it is observed that the range is greater than the mean, which shows that there is a potential outlier. In this case, that is the fridge as it has a power consumption way bigger than other appliances.

#### Central Tendency for grouped data

Some times a set of observations may be too large that representing the way we have done in the previous examples might become noisy and unpresentable. In such situations, the data is organized and presented in tables and tally. These tables are sometimes called frequency tables.  The idea of the central tendency is essentially the same, but with a little tweak for clarity. The table below represents the distribution of ages of students in a class
 Age (x) Frequency (f) fx 3 7 21 5 19 95 6 13 78 4 11 44 Total n=50 =238
Mean

The mean of grouped data is calculated using the following steps Calculate the total number of ages 3*735*19+6*13+4*11 = 238 The total number of students is the sum of the frequency, n=7+19+13+11=50 Hence mean=238/50=4.76 Therefore, the average mean age of students in the class is 4.76 years

Median

To obtain the median, a note from the table that there are 50 observations, we have to arrange the ages in ascending order of magnitude

 Age (x) Frequency (f) fx 3 7 21 4 11 44 5 19 95 6 13 78 Total n=50 =238
Since there are 50 observations, the median would be the average of the 24th and 25th value. We can trace that from the frequency table. 7+11=18 this means that the first 18 values are 3 and 4; 18+19=37, the next 19 observations are 5s; this implies that the median age is 5.

Mode

This is simply the age with the highest frequency, which is age 5, with a frequency of 19.

Range

The range would simply be the highest age minus the lowest age, which is 6-3= 3 years.