The average weight of 6 students is 62 kg. If a 7th student weighs 52 kg, what is the new average weight of all 7 students?

Question

The average weight of 6 students is 62 kg.  If a 7th student weighs 52 kg, what is the new average weight of all 7 students?

Liam Burke · Accepted Answer

Compute total weight of 6 students first.

First, you must determine the combined weight of the six students: #6times62 = 372# kg. When the seventh student is added, the total weight will be #TW = 372+52 = 424# kg. Now find the average weight of these seven pupils: #AW = 424/ 7 = 60.6# kg. Say 61 kg, if you would.

Andrew Adams · Answer

The new mean will be

#mu_"new"=62+1/7(52-62)#
#color(white)(mu_"new")~~60.57#

G_Ozdilek's solution is excellent because it is very detailed. This explanation will show you how to get there quickly. As can be seen in the Answer section above, the quick formula is #mu_"new"=mu_"old"+(x_7-mu_"old")/7# Why is this effective? We have 6 observations whose mean is 62 kg. If we add a 7th observation, its value will shift the mean to some new balance point. The amount of that shift is #1/7#th of the difference between the new value and the old mean. This is because each of the 7 observations contributes #1/7#th of its value to the calculation of the mean: #mu_"new"=(x_1+x_2+...+x_7)/7# #color(white)(mu_"new")=1/7x_1+1/7x_2+...+1/7x_7# This is the source of the formula: #mu_"new"=(x_1+x_2+x_3+x_4+x_5+x_6+x_7)/7# #color(white)(mu_"new")=(6[1/6(x_1+x_2+x_3+x_4+x_5+x_6)]+x_7)/7# #color(white)(mu_"new")=(6[mu_"old"]+x_7)/7# #color(white)(mu_"new")=(7mu_"old"-mu_"old"+x_7)/7# #color(white)(mu_"new")=mu_"old"+(x_7-mu_"old")/7# This formula also generalizes to computing #mu_(n+1)#, the average of #(n+1)# observations, with only the old average #mu_n# and the new observation #x_(n+1):# #mu_(n+1)=mu_n+1/(n+1)(x_(n+1)-mu_n)# If it looks complicated, just remember it's the old mean #(mu_n)# shifted left (or right) by the amount of "tug" the new observation has, which is #1/(n+1)#th of the difference between the old mean and the additional observation: #1/(n+1)(x_(n+1)-mu_n)#.

Emery Adams · Answer

undefined To find the new average weight of all 7 students, we can calculate the total weight of all 7 students, including the weight of the 7th student, and then divide this total weight by the total number of students (7 in this case).

The total weight of the first 6 students is $6 \times 62 $ kg.

So, the total weight of all 7 students is $ 6 \times 62 + 52$ kg.

Now, we divide this total weight by the total number of students (7) to find the new average weight.

The new average weight of all 7 students is $\frac{(6 \times 62 + 52)}{7}$ kg.

Calculating this gives us the new average weight.