Understanding Patterns: Statistics in Class 11 Mathematics
A beginner's guide to median, mode and quartiles. This post should answer your basic questions about median, quartiles and mode and how to evaluate them. I am a math educator teaching mathematics to high school students and college students for the past 3 decades.
I can tell you exactly what you need to be studying to understand median, quartiles and mode. In this age of ChatGpt where AI takes over teaching as well, I cannot stress how important it is for you to avoid common mistakes which students make. So, that's me, adding a humane approach to learning math.
Median, Quartiles and Mode. How to calculate it?
Once upon a time, in a quiet village, nestled between hills and a river there lived a girl named Lily. She spent her afternoons exploring the woods, collecting stones and arranging them in sequences. Her favorite numbers were 9, 17 and 22.
One sunny morning, Lily heard the villagers talking about the annual Harvest Festival. One of the games would be a grand math contest where the sharpest minds in the village could compete. Lily wanted to take part in the contest.
She started forming a pattern with her stones, with 9, 17 and 22 at the center. She imagined a peaceful village with 22 houses along a winding road. The youngest child in the village was 9 years old. The carpenter who could mend anything was 17 years old.
She introduced the baker, the farmer and a teacher, each with their own age. Explaining the data, she said, the median is like the center dividing the data into two equal parts. Quartiles divide the data into 4 equal parts.
In her data, the first quartile was where the younger folks live. The third quartile was where the older folks lived and the 2nd quartile was the median. Finally, explaining the mode, she said the mode is the value which occurs the most.
In the math contest, the judges were impressed by Lily's creativity. Not only did Lily win the contest, she also left a lasting impression on the villagers. Most mathematical concepts can be made accessible through story telling.
In layman's language, a median is the middle value of a set of observations arranged in ascending or descending order. For a set of discrete values, you have to first arrange the data in ascending or descending order. Then depending on whether you have odd number or even number of values, you calculate the median.
My students generally find it easy to calculate the median for odd number of observations. Even number of values require a little more of work, but with a little help, they usually get it right. What happens for a frequency distribution?
Basically, you can divide frequency distributions into 2 types. One, is a simple frequency distribution without class intervals. Here, you first calculate the cumulative frequency. I usually recommend using the lesser than cumulative frequency.
I tell my students to divide n ( the sum of the frequencies) by 2 and then look at the frequency table and identify the median. For a grouped frequency distribution, there are 2 methods of calculating the median.
First, you draw, the cumulative frequency curve using the ogive and then using the ogive, you determine the median. As I tell my students, you need a high degree of accuracy here. Alternately, there is a formula to calculate the median which makes it very easy, provided you remember it right.
Quartiles are the values which divide the data into 4 equal parts
There are 4 quartiles, the lower quartile, the second quartile also called the median and the third quartile. The difference between the third and the first quartile is called the inter quartile range.
How will you calculate the quartiles. Again, for a set of discrete values, you arrange the data in ascending or descending order. You then calculate the quartiles using the formula. I make my students practice the case when you have a fractional value as well.
Again, for a simple frequency distribution, you calculate the cumulative frequency. Then using this you will calculate the quartiles. For a grouped frequency distribution with equal class intervals, you can either use the ogive or the formula for calculating.
In simple language, mode is the value which occurs the most frequently
Calculating the mode is one of the easiest things to do. My students just love calculating the mode for a discrete set of values. Just take the value which occurs most frequently. In this context, note that you can also have 2 modes, called a bimodal distribution.
Coming to a frequency distribution, if the frequency distribution is simple, just take the value with the highest frequency as the mode. For a grouped frequency distribution, there are two ways of calculating the mode. My students just love the graphical method, where you plot a histogram and then move on to evaluating the mode.
The second method of calculating the mode is using the formula which is again quite simple.
Keep in mind that mean deviation and standard deviation are two ways of measuring the dispersion of data. Mean Deviation basically gives you how far the data is distributed from the mean or median. So, in Mean Deviation problems, you need to calculate two types, mean deviation about the mean and mean deviation about the median.
Again, there are two types of problems. Firstly, we have a set of discrete observations . These are easier to calculate.
Then we have a frequency distribution which may be grouped or ungrouped. For problems of these type, you will use the formula and evaluate. Formulas are given in the images.
This topic is fairly easy and my students usually get it right provided they remember the correct formula. Mean deviation is not a very reliable method of measuring the deviation. Hence we have the standard deviation which is a more reliable and widely used method to evaluate the dispersion.
The mean of the squared deviations is called the variance and standard deviation is the square root of the variance. You calculate the standard deviation for a set of discrete values and for a frequency distribution. The formula for calculating the standard deviation for a set of discrete values is quite simple.
For calculating the standard deviation for a frequency distribution with equal class intervals, you can use three methods. First, you have the direct method. If the values are large, you have two methods to calculate the standard deviation, namely, the Shortcut method or the Deviation method and the Step Deviation method.
Note that these two methods are more or less similar, with a slight difference. As I tell my students, you are free to use any formula of your choice in the exams. But keep in mind, that if a particular question demands the use of a particular method, then you have to abide by the question.
How Online Tutoring helps!
I have given you a brief gist on how you should go about preparing Mean Deviation and Standard Deviation. If you still need some help or guidance, you can contact me for online tutoring. You would require around three sessions of an hour each to understand this topic.