the table below gives the distribution of village under different height from sea level in certain reign compute the mean height of the region by short-cut method. height(in meters): 200 600 1000 1400 1800 2200 number village: 142 265 560 271 89 16
A = Assumed mean in this case A = 1400
x = height
N= frequency
The table:
x , N , dx = ( x - A ) , N ∙ dx
200 , 142 , - 1200 , -170400
600 , 265 , - 800 , - 212000
1000 , 560 , - 400 , - 224000
1400 , 271, 0 , 0
1800 , 89 , 400 , 35600
2200 , 16 , 800 , 12800
--- , ΣN = 1343 , --- , Σ N∙dx = - 558000
x̄ = A + ΣN∙dx / ΣN
x̄ = 1400 + ( - 558000 ) / 1343 =
1400 - 558000 / 1343 =
984.5123 ≈ 985 m
To compute the mean height of the region using the shortcut method, you need to multiply each height by its corresponding frequency (number of villages) and then take the sum of these products. Finally, divide the total by the sum of the frequencies (total number of villages).
Let's label the given heights as x and the corresponding frequencies as f.
The given data can be written as:
x = [200, 600, 1000, 1400, 1800, 2200]
f = [142, 265, 560, 271, 89, 16]
To calculate the mean height using the shortcut method, follow these steps:
1. Multiply each height by its corresponding frequency:
x * f = [200*142, 600*265, 1000*560, 1400*271, 1800*89, 2200*16]
= [28400, 159000, 560000, 379400, 160200, 35200]
2. Take the sum of the products computed in step 1:
Sum(x * f) = 28400 + 159000 + 560000 + 379400 + 160200 + 35200
= 1329000
3. Calculate the sum of the frequencies:
Sum(f) = 142 + 265 + 560 + 271 + 89 + 16
= 1343
4. Divide the sum of the products (from step 2) by the sum of the frequencies (from step 3) to get the mean height:
Mean height = Sum(x * f) / Sum(f)
= 1329000 / 1343
≈ 990.49 meters
Therefore, the mean height of the region, computed using the shortcut method, is approximately 990.49 meters.