Question

The following is a small data set of 25 observations. I want you to calculate some statistics by hand to cement the class material. You can use the regular formula for the variance or the computational formula - I’m just interested in the correct result. If you use the computational formula, you will want to calculate a separate row of each value squared in order to calculate the Sum(x^2). Excel can be used to solve this problem.

VarX
44
34
49
58
17
44
34
49
45
20
26
40
35
37
57
45
70
35
37
42
44
45
27
46
63

1. Create a Stem and Leaf Plot of the data (it is easy to do in Word by making a two column table and entering the numbers in there, bolding the stems)

b. Calculate the following (show your work):

• Mean
  • Median
  • Mode
  • Range
  • Variance
  • Std. Deviation
  • Coefficient of Variation

c.   Calculate a z-score for a data value of 70. Describe this in words.

Answer 1

We create a Stem and Leaf Plot of the above data is as below

Stem	leaf
1	7
2	0, 6, 7
3	4, 4, 5, 5, 7, 7
4	0, 2, 4, 4, 4, 5, 5, 5, 6, 9, 9
5	7, 8
6	3
7	0

We prepare the following table for finding above values

	Ascending Xi
44	17	41.72	5.1984
34	20	41.72	59.5984
49	26	41.72	52.9984
58	27	41.72	265.0384
17	34	41.72	611.0784
44	34	41.72	5.1984
34	35	41.72	59.5984
49	35	41.72	52.9984
45	37	41.72	10.7584
20	37	41.72	471.7584
26	40	41.72	247.1184
40	42	41.72	2.9584
35	44	41.72	45.1584
37	44	41.72	22.2784
57	44	41.72	233.4784
45	45	41.72	10.7584
70	45	41.72	799.7584
35	45	41.72	45.1584
37	46	41.72	22.2784
42	49	41.72	0.0784
44	49	41.72	5.1984
45	57	41.72	10.7584
27	58	41.72	216.6784
46	63	41.72	18.3184
63	70	41.72	452.8384
1043			3727.04

We find the following values using above table

1) Mean :- We find the mean of above data using following formula

Mean =

2) Median :- We find the median of above data using following formula

N = total number of observations = 25

For finding median, we arrange the data in ascending order which is in second column of above table.

Using second column of above table we find median

Median = value of (N + 1)/2 th observation

= value of (25 + 1)/2 th observation

= value of 13th observation

Median = 44

3) Mode :- We find the mode of above data using following way

Mode = Most frequent or repeated value in the data

Here, there are two mode exist because the data values 44 and 45 occuring in 3 times in the above data

the above data is bimodal

Mode = 44 and 45

4) Range :- For finding the range, we first arrange the data in ascending order which is in second column of above table.

Using second column of above table we find the range

The formula of range is

Range = Largest value - Smallest value

For above data,   Largest value = 70 and Smallest value = 17

Range = Largest value - Smallest value

= 70 - 17

Range = 53

5) Variance ():- We find the variance of above data using following formula

  

Variance =

6) Std. Deviation () :- We find the Std. Deviation of above data using following formula

  

  

(Round answer up to 4 decimal places)

Std. Deviation =

7) Coefficient of Variation (C.V.) :- We find the Coefficient of Variation of above data using following formula

  

(Round answer up to 4 decimal places)

Coefficient of Variation =

c. We calculate a z-score for a data value of 70

X = a data value = 70

We find z-score using following formula

Using above values of and we get z-score is as below

  

(Round answer up to 2 decimal places)

The above z-score is 2.32 means that the above value 70 is far away 2.32 standard deviation from mean which is equal 41.72.