Question

Here is a sample dataset: 126.3 258.9 298.4 345.2 89.2 459.2 688.1 441.3 459.2 487.1 647.2 784.9 647.2 675.5

(1) Draw a stem - and - leaf display for this dataset.

(2) What is the mean, median, mode?

(3) What is the range, variance, standard deviation?

(4) What is Q1, Q3?

(5) Is the smallest and largest outliers? Why?

(6) What is the percentage falling within 1 standard deviation? Does it satisfy the Empirical Rule?

Answer 1

1)

stem =100
leaf=10
stem	leaf
0	8
1	2
2	5 9
3	4
4	4 5 5 8
6	4 4 7 8
7	8

2)

mean =    ΣX/n =    6407.700   /   14   =   457.6929

Median=0.5(n+1)th value =    7.5th   value of sorted data
=   459.2  

mode= highest frequency data = 459.2 , 647.2

3)

range=max-min =    784.9   -   89.2   =   695.700
sample variance =    Σ(X - X̄)²/(n-1)=   603711.5493   /   13   =   46439.350
                      
sample std dev =   √ [ Σ(X - X̄)²/(n-1)] =   √   (603711.5493/13)   =       215.498

4)

quartile , Q1 = 0.25(n+1)th value=   3.75th   value of sorted data
=   288.525  
      
Quartile , Q3 = 0.75(n+1)th value=   11.25th   value of sorted data
=   654.275  

5)

IQR = Q3-Q1 =    365.8
  
1.5IQR =    548.63
  
lower bound=Q1-1.5IQR=   -260.10
  
upper bound=Q3+1.5IQR=   1202.90
  
outlier =values outside lower bound and upper bound  

there is no outlier

6)

X̄ ± 1 * s = (   242.19   ,   673.19   )

percentage  falling within 1 standard deviation=9/14 = 64.29%

according to Empirical rule , 68% of data values lies within 1 std dev away from mean,

So, it approximately satisfies the Empirical rule.