Question

A player keeps track of all the rolls of a pair of dice when playing a board game and obtains the following data. Find the mean, the median, and the mode

Answer 1

a) Mean = 7

b) Median = 7

c) Mode = 7

Complete Question:

The image of the data table is attached.

Explanation:

Mean, median and mode are all measures of central tendency.

Mean is the average of all the data generated that means addition of all the data divided by the number of data.

X = sum of all data/ total number of data

Median is the middle data, arranged in ascending or descending order.

Mode is the data with the highest frequency or occurrence.

a) Mean:

Sum of all the data S is

S = 2(10)+3(29)+4(40)+5(56)+6(68)+7(77)+8(67)+9(55)+10(39)+11(28)+12(11)

S = 3355

Number of data N = 10+29+40+56+68+77+67+55+39+28+11

N = 480

Mean = S/N = 3355/480

Mean = 6.99≈7

b) Median:

The median is the sum of the 240th and 241th data divided by 2 (since the total frequency is even)

Median = (7+7)/2 = 7

c) Mode

The number with the highest frequency.

The highest frequency is 77, therefore the mode is 7

Mode = 7

Answer 2

A player keeps track of all the rolls of a pair of dice when playing a board game and obtains the following data.

(a). X: 2, 3, 4, 5, 6, 7 (b). X: 8, 9, 10, 11, 12

F: 10, 29, 40, 56, 68, 77 F: 67, 55, 39, 28, 11

Find the mean, the median, and the mode

Answer:

(a.) Mean = 5, Median = 6, Mode = 9

(b). Mean = 9 , Median = 7, Mode = 8

Explanation:

Mean

The mean is the average value in a data set. To calculate the mean we use the following formula:

Sum (X)/n where Sum X is the sum of the values in the data, and n is the total number of data values.

For data in a frequency distribution such as the one in the above question we use the formula:

Sum (XF)/Sum(F) where F is the frequency of the values.

We calculate the mean as follows:

(a). Sum (XF) = (2x10) + (3x29) + (4x40) + (5x56) + (6x68) + (7x77) = 1494

Sum (F) = 10 + 29 + 40 + 56 + 68 + 77 = 280

Mean = Sum (XF)/Sum(F) = 1494/280 = 5.3 which is approximately 5 ( to the nearest whole number)

(b). Sum (XF) = (8x67) + (9x55) + (10x39) + (11x28) + (12x11) = 1861

Sum (F) = 67+ 55 + 39 + 28 + 11 = 200

Mean = Sum (XF)/Sum(F) = 1861/200 = 9.3 which is approximately 9 ( to the nearest whole number)

Median

The median is the central value of the data and is determined as follows:

First arrange the values in a data set in an ascending order (from the lowest to the highest)

Let n be the number of data points (values) in a set of data, then the median of the data is:

The value in the (n/2)th position, if n is an odd number

The value in the ((n+1)/2 )th position, if n is an even number

For data arranged in a frequency table as in the above case, to get the total number of values (n) in each data set, we sum up the frequency.

(a) n = 10 + 29 + 40 + 56 + 68 + 77 = 280 ( n is even)

n/2 =280/2 =140

the median is the value in the 140th position.

You can easily located the number in the 140th position by adding each value of the frequency starting from the top of the table until you obtain a value that greater or equal to 140:

10 + 29 =39,

39 + 40 =79,

79 + 56 =135,

135 + 68 = 203

We obtained a value greater or equal to 140 at the position of frequency 68, which means the median is the value in the position with frequency 68.

The median is 6

(b.) n= 67+ 55 + 39 + 28 + 11 = 200

n/2 = 100

The Median is the value in the 100th position, which is the position with frequency 55, using the same procedure in (a).

The median is 9

Mode:

The mode of a data set is the number that occurs most frequently

For a frequency distribution. The mode is simply the number with the highest frequency value.

Applying this to the above question:

(a.) The highest frequency value is 77, and the number with that frequency is 7, hence the mode is 7

(b). The highest frequency value is 67, and the number with that frequency is 8, hence the mode is 8