statistic Gurus..10points!?

Question

Miles/ Hour Frequency
50 - 54 2
55 - 59 4
60 - 64 5
65 - 69 10
70 - 74 9
75 - 79 5

what is the mean? Variance? Standard Deviation?

Answer 1

Mean (estimate)= (2((50+54)/2)+4((55+59)/2)
+5((60+64)/2)
+10((65+69)/2)
+9((70+74)/2)+5((75+79)/2))/(2+4+5+10+9+5)
=67

Variance (estimate)=(2((((50+54)/2)-67)^2)
+4((((55+59)/2)-67)^2)+5((((60+64)/2)-67)^2)
+10((((65+69)/2)-67)^2)+9((((70+74)/2)-67)^2)
+5((((75+79)/2)-67)^2)/(2+4+5+10+9+5)
=48+4/7 or 48.57(to 2 decimal places)

Standard Deviation (estimate)=
sqrt(Variance)
=sqrt(48+4/7)
=6.97 (to 2 decimal places)

NB: These figures are only estimates as this is grouped data and I am assuming that every value is the midpoint of its group.

Please pick me for best answer because this took quite a while to type!!!

Also, I have not just given you the formula but actually the right answers!!!

Answer 2

For the mean:
Compute the midpoint of each interval, multiply that times the number of its occurrences, then add all the numbers together and divide by the sum of the frequencies.
sum of frequencies = 2+4+5+10+9+5 = 35, so you divide by 35.
For the first interval the midpoint is
(50+54)/2 = 104/2 = 52
Multiply 52 by its frequency, 52*2 = 104
Now do that for the remaining 5 intervals.
Add the 6 numbers together.
Divide that by 35
The variance is the sum of the squares of the differences of the values and the mean, all divided by (35-1)
Use the midpoint for the values.
Say the mean is M, then the variance is
[(52-M)^2 + (57-M)^2 + (62-M)^2 + (the remaining three intervals)] /34
The standard deviation is the squareroot of the variance.

Answer 3

Since we are not given a continuous function to intergrate, I will treat this as a discrete function using the average of each of the ranges. Therefore the data becomes:
average*count
52 mph * 2 =104
57 mph * 4 =228
62 mph * 5 =310
67 mph *10 =670
72 mph * 9 =648
77 mph *5 = 385
The total number of counts = 35
sum is 2345, sample mean = 2345/35 = 67, that looks right.

Sum (value - mean)^2*(count/total) = variance

(52 -67)^2*(2/35) + (57-67)^2*(4/35) + (62-67)^2*(5/35) +
(67-67)^2*(10/35) + (72-67)^2*(9/35) + (77-67)^2*(5/35) = var

Whatever the variance sums to, take the square root of that to get the Standard Deviation.

Answer 4

Mean = (each value x no of its occurances) / total occurances. In your example, (50*54.2+55*59.4+,,,+ 75*79.5)/(54.2+59.4+...+79.5)

Variance is the sum of the squares of each value's occurrance minus the mean; if the mean M was calculated above, (50- M)^2 * 50 is the contribution to variance from the 50mph samples. Do this for each sample and add.

Standard deviation is the square root of [variance / (no of samples - 1)].

Answer 5

Mean= 67
Variance= 4538
Standard Deviation=67.36