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
Step-by-step 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
