Question

MODEL ONE: Lengths of Mango Fruit

The table shows distribution of lengths of a sample of mangos.

LENGTHS (cm)	FREQUENCY
12.5 – 16.4	3
16.5 – 20.4	6
20.5 – 24.4	7
24.5 – 28.4	9
28.5 – 32.4	4
32.5 – 36.4	2

1. In which range is the median mango length? Explain.

2. Calculate the mean length of the mangos in the sample. Round to the nearest tenth of a centimeter.

3. Suppose that the two longest mangos were measured incorrectly, and that they were both slightly longer.
   1. Why would the median and quartiles not change?

2. Would the mean remain the same, increase, or decrease? Why? (Explain without doing any calculations.)

3. Would the standard deviation remain the same, increase, or decrease? Why? (Again, please explain without doing any calculations.)

Answer 1

The given table is the distribution of the lengths of mangoes.

Now, the values of the class boundaries are given. We have to convert them into the class limits.

The frequencies are given for each class. So, for the mean we need to calculate the mid values of the classes, and for the median class, we have to calculate the cumulative frequencies of the classes.

So, the final table is

LENGTHS (Cm)	LENGTH BOUNDARIES	FREQUENCY	CUMULATIVE FREQUENCY	MIDVALUE
12.5-16.4	12.45-16.45	3	3	14.45
16.5-20.4	16.45-20.45	6	9	18.45
20.5-24.4	20.45-24.45	7	16	22.45
24.5-28.4	24.45-28.45	9	25	26.45
28.5-32.4	28.45-32.45	4	29	30.45
32.5-36.4	32.45-36.45	2	31	34.45

Now, we know that the mean length is defined as

So, the mean length of the mangoes is approximately 23.9 cm.

Now, here the total frequency is 3+6+7+9+4+2, ie. 31.

So, n = 31.

Now, the integer part of half of n, is 15.

Now, the cumulative frequency of the class 20.5-24.4, is 16, which just exceeds 15.

So, the median mango length is in the range of 20.5-24.4 cm.

Now, it is found that the lengths of the two of the longest mangoes were ,measured incorrectly; both of them are slighly longer.

(a) Now, the medians and the quartiles are the values dependent on the orderings of the values. Even when the longest two mangoes have slightly longer heights, they remain the longest, and their positions in the ordered arrangements, do not change.

So, the medain and quartiles do not change.

(b) Now, even when the lengths of the longest two mangoes are slightly more, they still remain within the same range of group, which is 32.5-36.4. So, they are stll represented by the same midvalue, ie. 34.45.

So, the mean also does not change.

(c) Now, even when the lengths of the longest two mangoes are slightly more, they still remain within the same range of group, which is 32.5-36.4. So, they are stll represented by the same midvalue, ie. 34.45.

So, the standard deviation also does not change.

Note: The mean and standard deviation does not change here because the data is grouped, and the data in each group is replaced by a single midvalue, which is not subject to change. If the data was raw or ungrouped, the mean and standard deviation would increase, with the longest two mangoes being slightly longer.