Arithmetic Mean | Geometric Mean | Harmonic Mean

0:00 Introduction
1:21 Arithmetic mean
3:25 Geometric mean
8:59 Harmonic mean
14:29 Challenge Question

[martinplasse]

If you can't explain something simply and clearly you don't truly understand it. This guy truly understands statistics. A lot people, including most teachers I would venture, have become proficient at statistics without a genuine understanding of what they are doing. Thx for this.

[vuhuynh]

I'm two years too late to the party. But I'd like to put up my answer anyway. A superb video btw. Really appreciate it.
I'll do it in 2 steps.
Step 1: Find the average speed for stage 1.
Since the two activities (the climb and the descent) are of the same distance but at different speeds, the average speed for stage 1 must be the harmonic mean of two speeds.
Average speed (stage 1) = 2 / (1/10 + 1/50) = 16.67 km/h.

Step 2: Find the average speed between stage 1 and stage 2.
Since stage 1 and stage 2 involve the same amount of time (both are 30 minutes long), the average speed would be the arithmetic mean.
Average speed (stage 1 & 2) = (16.67 + 40) / 2 = 28.33 km/h.

I'm sure others already came up with the perfect answer long ago. Still, I wanted to try myself.
Thanks a ton for the examples. They really put all the different types of means in perspective.

[tsvianshell]

You don't have to use harmonic mean, a weighted mean is also possible:
On stage one, since uphill distance equals downhill distance (same hill) - the times spent on every part can be found using the equation:
10X=50(30-X) which gives X=25.
The next stage is just a weighted mean:
(10*25+50*5+40*30)/60=28.33

[Raudadrosin]

I really are the best statistic teacher EVER! How deep you go in explaining everything is finally making me understand... I´ve been so lost because I didn´t have the foundation for the understanding!
Both the lectures + how you talk and describe = perfect in every way!
I really hope that I will finally pass my statistics esam this Saturday!

[arashchitgar]

first stage: we have fixed distance, so harmonic mean for km/h:
(1/10)+(1/20) = 16.66

1stage + 2nd stage: we have a fixed time, so arithmetic mean for km/h:

(16.66 + 40) / 2 = 28.33

[rohit-rvcr]

We will find harmonic mean of two activities of stage one and then we will find arithmetic mean for both the stages.

so, harmonic mean of stage one = 2/(1/10+1/50)= 16.67. Now the arithmetic mean for both the stages = (16.67+40)/2=28.33

His average speed is 28.33 Km/h.

btw, great video i was confused why we use geo mean. You were excellent.

[denki]

Stage 1 - HM = 2/(1/10+1/50) = 50/3 ≈ 16.66
Overall Average - AM = (HM + 40) / 2 = 20 + 25/3 = 85/3 ≈ 28.33
units are in kmph

Edit for checking:
let x = time taken to ride uphill in hours
since downhill took 5 times faster than uphill, then 0.5-x=x/5 -> x=25/60 (25mins)
Total Distance = 10x + 50(0.5-x) + 40*0.5 = 25/6 + 25/6 + 20 = 85/3 ≈ 28.33
Total Time = 0.5 + 0.5 = 1 (1 hour)
Average Speed ≈ 28.33 / 1 ≈ 28.33

[pfever]

Challenge question: What is the average speed over the whole track? Ans. 28.33 km/h
Stage 1.- 30 mins - up-hill 10km/h, down-hill 40km/h, which means, equal distance traveled in a different time, therefore use harmonic mean.
Stage 2.- 30 mins - 40km/h, equal time as stage 1, so use the arithmetic mean between them.

average of stage 1: ( (1/10 + 1/50) / 2 )^-1 = 16.66 km/h
average of stage 1 and stage 2: (16.66 + 40) / 2 = 28.33 km/h

[cannaroe]

Such a great video! Square-rooting finally clicked for me, because I always think of it as returning a single number, but it's actually returning 2 numbers, or 3 for cube-root, but because all the numbers/dimensions are the same we just refer to the product as a single number. Applying that to ROI was great! Maybe at 14:04 it would help some people to know 1 Hertz is 1 thing per second, so that's where the ratio is coming from, the fact that you are talking about 2 dimensional frequencies, oscillations in time, but referring to it as a single number. Harmonic was explained so well i wish i could subscribe twice!

[seekingalpha_]

i cant thank you enough for this series. Your explanation is literally so on point it couldn't be anymore perfect than this!

[Anukul]

In the first part, the mean is across 2 equal distances (n = 2). So, ((1/10)(1/50)/2)^-1 = 16.67
The mean along the first and second part is across equal time, so it is not GM but AM so (16.67+40)/2 = 28.33

[endubi]

The best statistics teacher ever! You break down complex topics into simple terms that can easily be understood.

[tatiana]

Brilliant! Thank you so much. I felt dumb because I was unable to solve the final question but I understood the solution in the comments.

[hhhhhhahah]

Harmonic Mean for Stage One = 16.6km/h, Stage two mean is fix to 40km/h. I then computed the arithmetic mean: *28.3 km/h* is this correct?

[AJ-fohp]

Thank You for your explanation
Simple example, simple sentences, crystal clear.
There is one more is there Root Mean Square Value(RMS)

[christopherlinder]

Great video. Short enough for an intro, but long enough for some very helpful details. I came across your video after recently having seen a question along the lines of "Ben takes 2 hours and 12 minutes to dig a hole, Bob beats him and digs the same hole in 1 hour and 50 minutes, how long would they have taken to dig the hole together in joint effort?" which involves basically half of the harmonic mean, in this case, and that got me interested. It also reminded me of the fact that runners often use "pace", which is min/km, rather than, say speed in km/h. So a pace of 6 would be 10 km/h, and a pace of 8 would be 7.5 km/h. Essentially, since one is having to take the reciprocal in order to add the speeds before one flips it back into pace with another reciprocal, these types of questions of "resultative times in combined effort" are related to the harmonic mean. Oddly though, I have to remark that with musical notes and their frequencies, at least with well-tempered tuning like the piano, where let us say A = 440 Hz and the frequencies of each semitone higher is the precursor frequency multiplied by 12th root of 2 which is around 1.0595 or 5.95% higher, this entails that finding the frequency of the notes in between two notes involves the geometric mean. So whereas the arithmetic mean of the 440 Hz A and the 880 Hz A would be an E of 660, which is "too high" because it is a natural fifth away from the low A and only a natural fourth away from the high A, nicely shown by it being 7 semitones away from the low note and 5 semitones away from the high note, if you want the D# or E flat that's 6 semitones away (a tritone) from both, you can take the geometric mean of sqrt( 440 x 880 ) and you will get exactly 440 times sqrt(2), which is around 622.25 Hz. (Of course, with waves and music being what they are, A-E-A (Strauss's Zarathustra) will always sound great, and like "the middle" because it's consonant and A - D# - A (Holst's Planets, Mars) will always sound scary because its a tritone and dissonant, but such is the odd design of the keyboard, and tempered tuning is mathematically always a "compromise" of sorts.

[divine]

Correct me if I'm wrong, but isnt the challenge problem flawed? In stage one, we cannot assume the distance uphill equals the distance downhill- we know only that the sum of the time taken to go uphill is 0.5hrs ie: t1+t2=0.5, and that 10t1 + 50(0.5-t1)= total distance travelled in stage one denoted by D1. There's no way to link any component of D1 to other bits of information given. Therefore, there are 2 degree of freedom (ie: (t1 or t2) and a component of D1): there's no unique solution.

[radhakrishntextile]

Thankyou for bringing conceptual clarity

[asifhasan]

30 min @10km/hr. That’s 5 km in 30 min.
5km back @50km/hr. 5 km in 6 min.
30 min @40km/hr. 20 km in 30 min.
Total distance 30 km
Total time 66 min

Avg speed (30/66)*60=27.272727.... Km/hr

[TheScawer]

First time I really understand harmonic mean, thank you! Your way of inverting it (twice) really helps with the understanding.