Above image shows some data related to a walk. It says that the person has walked a distance of 7.05 km in 1 hour 8 minutes 49 seconds. Average pace shown here is 9 minutes 45 seconds per Km and the time taken in different kilometers are given just below that. This average pace can be simply computed by dividing the total time (1 hour 8 minutes 49 seconds) by the distance walked (7.05 Km) and doing the required conversions between hour, minutes and seconds. This means, if I walk each of the 7 Kms at the same pace completing them in exactly 9 minutes 45 seconds then the total time taken by me to walk 7.05 Kms would be 1 hour 8 minutes 49 seconds. In the last .05 Km too, the pace has to be same.

Time taken in each of the 7 Kms are given in the image. Let us check, whether the walking pace was higher or lower than the given average in the last .05 Km. Before reading further, pause a bit and think about the process you would like to follow to do this.

I plan to check whether the given average 9 minutes 45 seconds is same as the average of time taken in 7 Kilometers. If yes, then the pace was same as the average in last .05 Km after the 7 Kms. Otherwise, we will draw conclusion based on whichever average is higher.

Time taken in walking different kms are given as 10'48", 9'34", 9'29", 9'42", 9'30", 9'26" and 9'43". One can add them and divide by 7 to get the average. But we make a guess or just take 9'45" as the average and check how much more or less it is of each of the given data. If the sum of these differences is 0, the average is 9'45" otherwise we make necessary correction in the assumed average. So, 10'48" is 63" higher than the assumed average 9'45". We take this +63". The second data 9'34" is 11" lower than the assumed average. We note -11. We do the same for rest of the data and get -16", -3", -15", -19" and -2". Add all these to get -3". So, the average of these times taken in completing 7 different kilometers is less than 9'45" by 3/7 seconds. Since, the time taken in the additional 0.05 Km is increasing the average pace (time/km), the walking speed in this part is slower than the average or time taken/km is higher than the average.

Here, the term pace is being used to talk about walking pace and it has a unit time/km. It may be confusing to see that when you increase the walking pace (you think so by walking faster), the time/km decreases. Many of us might be used to walking speed measured in distance per unit time, like 6 Km/hr. or 4 miles/hr. or 3 meters/sec etc. That provides the comfort that when you increase the walking speed, the number showing the speed also increases.

It is interesting to observe that the average behaves in somewhat different way when you use speed in terms of distance per unit time. So, if we are given that the first km is walked at a speed of 5 km/hr., 2nd at a speed of 6 km/hr., 3rd at 6.5 km/hr. 4th at 6 km/hr., 5th at 7 km/hr., 6th at 6km/hr. and 7th at 5.5 km/hr., the average speed in entire walking is not equal to sum of these 7 speeds divided by 7. When you are walking at slower speed, you are taking more time to walk one km compared to the time taken by you when you walk at faster speed. Average speed still remains distance/time, distance remains same in all the 7 kms but the time which varies due to speed is in the denominator. If you know arithmetic mean and harmonic mean, you should be able to relate that here harmonic mean concept needs to be applied whereas in the data given in the image (time/distance) above arithmetic mean is to be used.

It's important to understand the concepts clearly here to avoid serious problems at later stage. Average is a very popular analysis people do with the data, and many times they base their preferences on this kind of computation. Two situations are there in this article. One is walking pace given as time taken per kilometer and the other one is kilometers walked per unit time. Compute the average in both cases, check the process, think what the computation represents at different stages and get clear idea where harmonic mean needs to be used instead of arithmetic mean.

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