Answer to Question #100671 in Statistics and Probability for Low Zhi Lok

Question

Collect a set of grouped data and state the types of variable.
(i) Based on your grouped data, construct a histogram.
(ii) Estimate the mode from your histogram and interpret the mode value.
(iii) Calculate the mean and the standard deviation.

Answer 1

Length of 50 baby carrots (population) grown in a garden. The length of carrots are grouped into 8 classes and the number of carrots in each class (frequency) recorded.The above grouped data fall into the interval categorical variable since it involves numeric scales in which with known order and exact differences between the values.

(i). To construct a histogram, class boundaries are necessary since the values are not continuous.

Below is the R code used to generate the Histogram

x<-c(5,2,6,8,9,11,6,3)
names(x)<-c('149.5-154.5','154.5-159.5','159.5-164.5','164.5-169.5','169.5-174.5','174.5-179.5','179.5-184.5','184.5-189.5')
barplot(x,space=0,xlab='Lenght (mm)',ylab='Frequency',col=1:8)

(ii). It is clear from the histogram that the mode lies in 175-179 class. Mode is the peak of data set so, the mid point of the tallest histogram bar (class) is the mode. Mid point $=\frac{175+179}{2}=177$. Therefore, 177 is the value that occurs most frequently in the set of observations.

(iii). To obtain mean, mid point values (x) and fx values, which are the product of the midpoints and the frequencies need to be obtained.

$Mean=µ=\frac{Ʃfx}{Ʃf}=\frac{8530}{50}=170.6 mm$, f is the frequency of classes.

Standard deviation is given by $s=\sqrt{\frac{Ʃfx^2}{Ʃf}-µ^2}$

$s=\sqrt{\frac{1459920}{50}-170.6^2}=\sqrt{29198.4-29104.36}$

$=\sqrt{94.04}=9.7$