Question

The following data shows the different sizes of nails contained in a box bought by a
customer.


length(mm) Number of nails
10 to 14 10
15 to 19 12
20 to 24 18
25 to 29 16
30 to 34 4

(a) Find the followings:
i. Mean

ii. Median

iii. Standard deviation

(b) Calculate the Pearson’s coefficient of skewness and explain the distribution.

(c) Construct a cumulative frequency curve. From the cumulative frequency curve,
determine:

i. The first quartile

ii. The third quartile

iii. The percentage of nails with length exceeding 22mm.

Answer 1

Answer on Question #63439 – Math – Statistics and Probability

The following data shows the different sizes of nails contained in a box bought by a customer.

Question

**(a)** Find the followings:

i. Mean

ii. Median

iii. Standard deviation

Solution

**(a)**

i. Mean

The "midpoint" of each class can be calculated as: Midpoint = (Lower class limit + Upper class limit) / 2 .

$$\bar{x} = \frac{1}{f} * \sum f_i * x_i$$

Approximate mean = $\bar{x} = \frac{(12*10+17*12+22*18+27*16+32*4)}{60} = 21,33$

The mean is 21.33 mm

ii. Median

There are 60 in this data set. The 23th to 40th distances all lie in class interval [20–24], and so the median also lies this class interval.

Now, the median is the $\frac{60+1}{2} = 30.5^{\text{th}}$ score

Median = $20 + \frac{30.5 - 22}{18} * 3 = 21.42$

The median is 21.42 mm.

iii. Standard deviation

$$\bar {x} = 2 1, 3 3$$

Standard deviation $= s = \sqrt{\frac{f(x - \bar{x})^2}{n - 1}} = \sqrt{\frac{2073.32}{60 - 1}} \approx 5.93$ .

Standard deviation is 5.93.

Question

(b) Calculate the Pearson's coefficient of skewness and explain the distribution.

Solution

Pearson's Coefficient of Skewness uses the median. The formula is:

$$S k _ {2} = \frac {3 (\bar {x} - M d)}{s}$$ $$\bar {x} = 2 1, 3 3$$ $$s = 5. 9 3$$ $$M d = 2 1. 4 2$$

Pearson's coefficient of skewness $= Sk_{2} = \frac{(\bar{x} - Md)}{s} = \frac{3*(21.33 - 21.42)}{5.93} = -0.045$

Therefore the distribution is skewed left

Question

(c) Construct a cumulative frequency curve. From the cumulative frequency curve, determine:

i. The first quartile

ii. The third quartile

iii. The percentage of nails with length exceeding $22\mathrm{mm}$ .

Solution

(c)

We need to add a class with 0 frequency before the first class and then find the upper boundary for each class interval.

And then plot the cumulative frequency against the upper class boundary of each interval and join the points with a smooth curve.

From the cumulative frequency curve, determine:

According to curve:

i. The first quartile

$$Q1 = 21.11$$

ii. The third quartile

$$Q3 = 26.56$$

iii. The percentage of nails with length exceeding $22\mathrm{mm}$.

The percentage of nails with length exceeding $22\mathrm{mm} = 51.67\%$.

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