Question

Andrew asked seven of his friends how many cousins they had. The results are listed below. 15 12 514446a. Find the mean, median, range, midrange, and standard deviation of the sample data. Round to tenths. b. Find the 5-number summary and construct a boxplot.

Answer 1

a) Mean = 8.6; Median = 6; Range = 4 to 15; Mid-range = 9.5; Standard deviation = 4.5

b) the sample minimum - 4; the first quartile - 4; the median - 6; the third quartile - 14; the sample maximum - 15

Box plot attached

Explanation:

As the question doesn't clearly mention the response of each of Andrew's friends, it is assumed as 15, 12, 5, 14, 4, 4, 6 as there should be 7 responses

a)

Step 1: Find the mean

Mean is the value obtained by dividing the sum of several quantities by their number

Mean = (sum of all values)/(number of values)

Mean = (15+12+5+14+4+4+6)/(7)

Mean = 60/7

Mean = 8.57

Mean = 8.6

Step 2: Find the median

Median denotes the value or quantity at the midpoint of a frequency distribution of values or quantities, meaning there is an equal probability of a value in the range being above or below it

Ordered distribution : 4, 4, 5, 6, 12, 14, 15

Median = 6

Step 3: Find the range

The range is the area of variation between upper and lower limits on a particular scale

Range = 4 to 15

Step 4: Find the mid-range

Mid-range is the arithmetic mean of the largest and the smallest values in a sample

Mid-range = (lowest value + highest value)/2

Mid-range = (4 + 15)/2

Mid-range = (19)/2

Mid-range = 9.5

Step 5: Find the standard deviation

Standard deviation is a value expressing by how much members of a group differ from the mean value for the group. It is calculated by taking the square root of the sum of the squares of the difference between each value and the mean divided by the number of values

Standard deviation = sqrt{sum[(mean - value)^2]/number of values}

Standard deviation = sqrt{[(8.57-4)^2 + (8.57-4)^2 + (8.57-5)^2 + (8.57-6)^2 + (8.57-12)^2 + (8.57-14)^2 + (8.57-15)^2]/7}

Standard deviation = sqrt{[(4.57)^2 + (4.57)^2 + (3.57)^2 + (2.57)^2 + (-3.43)^2 + (-5.43)^2 + (-6.43)^2)]/7}

Standard deviation = sqrt{[20.885 + 20.885 + 12.745 + 6.605 + 11.765 + 29.485 + 41.345]/7}

Standard deviation = sqrt{143.715/7}

Standard deviation = sqrt{20.531}

Standard deviation = 4.53

Standard deviation = 4.5

b)

The five number summary is a set of descriptive statistics that provides information about a set of values:

1. the sample minimum - 4
2. the first quartile - 4
3. the median - 6
4. the third quartile - 14
5. the sample maximum - 15

The box plot showing these values is attached

Answer 2

Mean = 33.6, Median = 44, Range = 39, Midrange = 31.5, σ = 16.59.

Explanation:

a) First, we will find the mean of the sample data;

Mean = 15+12+51+44+46 / 5

Mean = 33.6.

Second, we will find the median of the sample data;

for median you have to write your data set when in order from least to greatest then, the median is the middle number of the set:

12, 15, 44, 46, 51

Median = 44.

Third, we will find the range of the sample data;

Range is the difference between the highest and lowest values:

Range = 51 - 12

Range = 39.

Fourth, we will find the midrange of the sample data;

To find the midrange, add together the least and greatest values and divide by two, or in other words:

Midrange = 51+12 / 2

Midrange = 31.5.

Fifth, we will find the standard deviation of the sample data;

The formula of standard deviation is:

σ = √1/N Σi=1 to N (xi - mean)^2

now, subtract the mean and square the result

(15 - 33.6)^2 = 345.96

(12 - 33.6)^2 = 466.56

(51 - 33.6)^2 = 302.76

(44 - 33.6)^2 = 108.16

(46 - 33.6)^2 = 153.76

now, add them and divided by the number:

mean of squared differences = 345.96+466.56+302.76+108.16+153.76 / 5

mean of squared differences = 275.44

and Now, square root it, we will get standard deviation

σ = √275.44

σ = 16.59.

b) The 5-number summary;

minimum = 12

maximum = 51

median = 44

(12, 15,) 44, (46, 51)

Quartile1 = 13.5

Quartile3 = 48.5

Now, the summary is:

minimum = 12, Quartile1 = 13.5, median = 44, Quartile3 = 48.5, maximum = 51.