Describing Data

Question

Consider the following data set. Round your answers to the nearest hundredth as needed.

46	86	77	65	57
86	70	86	64	55

1. Mean = 
2. Median = 
3. Mode = 
4. Range = 
5. Sample Standard Deviation =

Answer 1

Mean is the average of all data elements in the dataset. Add all data elements together and divide by the number of data elements. 692/10 = 69.2

For Median, Mode and Range, I suggest putting the data elements in the dataset in numerical order:

46 55 57 64 65 70 77 86 86 86

Median is the middle data element in the dataset. With an odd number of data elements, the median is the middle number. With an even number of data elements, like this example, the median is the average of the two middle numbers. (65 + 70)/2 = 67.5

Mode is the most common data element in the dataset. Because the data elements are in numerical order, it is clear that 86 occurs 3 times, so Mode is 86

Range is the difference between the highest data element and the lowest data element. Because the data elements are in numerical order, it is clear that the highest data element is 86 and the lowest data element is 46, so the Range is 86-46 = 40

Standard Deviation is the measure used to define how close the data elements are to each other. It is this formula ::

where

s - the standard deviation

n - the number of data elements in the dataset,

×_i - which means the individual value of each data element

x with a bar over the top - which means the mean of the dataset

∑ - which means sum, i - the specific data element

√ which means square root and

( ) ² which means the quantity inside the parentheses is squared ^.

The whole process measures the difference of each data element with the mean of the dataset, squares the result and sums these results for all data elements. This number is then divided by one less the number of data elements in the dataset. Finally you take the square root of that number to get your standard deviation. Standard deviation = 14.30462241

It is far easier to use your calculator to do the heavy lifting of the calculation.

Answer 2

Mean 

69.2

Median 

67.5

Mode

86

Range

40

If you need any help understanding how to compute the above, please ask.

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Sample Standard Deviation, s: 14.304622407537

Count, N:	10
Sum, Σx:	692
Mean, x̄:	69.2
Variance, s2:	204.62222222222

Steps

s2 =
Σ(xi - x̄)2
Divided by: N - 1
Numerator computation:
= (46 - 69.2)2 + ... + (86 - 69.2)2
Denominator calculation:
10 - 1 = 9
The numerator sum is = 1841.6
Divided by 9
And the quotient is =	204.62222222222
s =	√204.62222222222
=	14.304622407537