New year, new you! You decide to change a few things about your lifestyle and generally make it healthier and more active. On the one hand, you choose to cut down on meat and eat more fruit and vegetables. On the other hand, you want to move around a little more and take up jogging.
The first few weeks were quite promising — you managed to stick to your plan and you already feel a lot better! You can even see that in the distance you can run before you feel tired and want to take a nap. It even encouraged you to begin writing down how much you jog each time on a piece of paper and then analyze your improvement.
The distances you wrote in the spreadsheet are:
1.9 mi, 1.7 mi, 2 mi, 2.3 mi, 1.8 mi, 2 mi, 2.4 mi, 2.1 mi, 2.4 mi.
But then you had a couple of weeks with a difficult project at work with too much overtime, and there was this great show on TV that glued you to the couch for many hours. As a result, the next few runs were not as impressive as the previous ones:
0.9 mi, 1.2 mi, 1 mi, 1.7 mi, 1.3 mi, 1.3 mi.
But you're not giving up! You're determined keep up the good work and, as the first step, you decide to reach the 60th percentile of the values before they went down. However, just to be realistic about your current stamina, you decided that before you get there, the next few runs should be around the 60th percentile of all the above values.
So, what is the distance you should jog the next time you go to the park? And what is the goal you want to reach in the not-so-distant future? With our percentile calculator, we'll know how to find the percentiles in no time!
First of all, let's see how to calculate the percentile of the smaller set of numbers using the percentile calculator. Observe that there are only eight fields to write the numbers. But don't you worry – once you fill all of them, more will appear!
We input the nine distances one by one as entries #1 to #9 into the percentile calculator. Lastly, at the bottom, we need to give the percentile we want to find (in our case, it's 60). Once we write that, the answer will pop up under the variable fields. That is the goal distance that we want to get to soon.
Next, we'd like to see how far we need to run the next time we go jogging. Note that we can find it by simply adding the six extra distances onto what we just input into the percentile calculator. So, we leave what's already there behind and simply add entries #10 to #15. Observe how the percentile changes with every number you give (as long as you still have the 60 in the Percentile field from before). And, the moment we input the last value, we get our final answer at the bottom.
Now, just to be sure, let's see how to calculate the percentiles ourselves to see how much time the Omni Calculator website can save us on a daily basis. So sit back at your desk, and let's get it over with!
We begin with the calculations for the smaller data set. The first thing we need to do when finding percentiles is to order the entries from smallest to largest. In our case, this changes the sequence from:
1.9 mi, 1.7 mi, 2 mi, 2.3 mi, 1.8 mi, 2 mi, 2.4 mi, 2.1 mi, 2.4 mi
to:
1.7 mi, 1.8 mi, 1.9 mi, 2 mi, 2 mi, 2.1 mi, 2.3 mi, 2.4 mi, 2.4 mi.
Now, we follow the steps from the Percentile formula section. This means that we first calculate the rank:
rank = (k / 100) × (n + 1) = (60 / 100) × (9 + 1) = 0.6 × 10 = 6,
where k is the percentile and n is the number of entries, which, in our case, is k = 60 and n = 9.
Next, we find the integer part and the fraction part:
integer_part = ⌊rank⌋ = ⌊6⌋ = 6;
fraction_part = rank - integer_part = 6 - 6 = 0.
According to the percentile formula:
kth_percentile = am + fraction_part × (am+1 - am),
this gives:
60th_percentile = a₆ + fraction_part × (a₇ - a₆) = 2.1 mi + 0 × (2.3 mi - 2.1 mi) = 2.1 mi.
So our goal distance is 2.1 mi. Well, looking at the last few numbers, we're not really there yet... Let's check out what we should start with.
Now, we do the same calculations but for a different, larger data set. Again, we begin with ordering it, so:
1.9 mi, 1.7 mi, 2 mi, 2.3 mi, 1.8 mi, 2 mi, 2.4 mi, 2.1 mi, 2.4 mi, 0.9 mi, 1.2 mi, 1 mi, 1.7 mi, 1.3 mi, 1.3 mi
becomes:
0.9 mi, 1 mi, 1.2 mi, 1.3 mi, 1.3 mi, 1.7 mi, 1.7 mi, 1.8 mi, 1.9 mi, 2 mi, 2 mi, 2.1 mi, 2.3 mi, 2.4 mi, 2.4 mi.
Next, we find the rank, the integer part, and the fraction part:
rank = (60 / 100) × (15 + 1) = 0.6 × 16 = 9.6;
integer_part = ⌊9.6⌋ = 9;
fraction_part = 9.6 - 9 = 0.6.
Lastly, we input the numbers into the percentile formula:
60th_percentile = a₉ + fraction_part × (a₁₀ - a₉) = 1.9 mi + 0.6 × (2 mi - 1.9 mi)
= 1.9 mi + 0.6 × 0.1 mi = 1.96 mi.
To sum it all up, we start by running 1.96 miles, but that's just a springboard to get us to 2.1 miles each time. Fair enough, seems like a good plan.
Encouraged and excited, you put on your running shoes and go out in the direction of the city park. You can already feel the endorphins kicking in and the calories being burned with each step. Sure enough, you'll have that beach body ready for summer in no time!
