a) Find the values of x and y so that the ordered data below has a median of 31 and a mean of 31.25.
13, 17, 25, 27, 28, x, 32, 36, 37, 40, y, 47
b) The numbers below have been written in ascending order. Find the values of a, b, c and d if the mode is 15, the median is 17, the range is 16 and the mean is 19.
13, a, 15, b, 18, d, c
a)
13 , 17 , 25 , 27 , 28 , x , 32 , 36 , 37 , 40 , y , 47
The median is the middle number.
But there is no "middle" number, because there are an even number of numbers. In this case, the median is the mean (the usual average) of the middle two values:
The median = ( x + 32 ) / 2
In this case median = 31 so:
( x + 32 ) / 2 = 31 Multiply both sides by 2
x + 32 = 31 * 2
x + 32 = 62 Subtract 32 to both sides
x + 32 - 32 = 62 - 32
x = 30
The mean is the usual average, so:
( 13 + 17 + 25 + 27 + 28 + x + 32 + 36 + 37 + 40 + y + 47) / 12 = 31.25
( 13 + 17 + 25 + 27 + 28 + 30 + 32 + 36 + 37 + 40 + y + 47) / 12 = 31.25
( 332 + y ) / 12 = 31.25 Multiply both sides by 12
332 + y = 31.25 * 12
332 + y = 375 Subtract 332 to both sides
332 + y - 332 = 375 - 332
y = 43
b)
13 , a , 15 , b , 18 , d , c
The range is the difference between the largest and smallest values.
In this case:
Range = c - 13
c - 13 = 16 add 13 to both sides
c - 13 + 13 = 16 + 13
c = 29
The mode is 15
The mode is the number that is repeated more often than any other, so mode = 15
this mean
a = 15
Becouse a is betwen 13 an 15.
The median is the middle number so:
median = b = 17
b = 17
mean = 19
( 13 + a + 15 + 17 + 18 + d + 29 ) / 7 = 19
( 13 + 15 + 15 + 17 + 18 + d + 29 ) / 7 = 19
( d + 107 ) / 7 = 19 Multiply both sides by 7
d + 107 = 19 * 7
d + 107 = 133 Subtract 107 to both sides
d + 107 - 107 = 133 - 107
d = 26
The numbers:
13 , 15 , 15 , 17 , 18 , 26 , 29