Respuesta :
Answer:
The range is 21
The interquartile range is 6
The variance is 33
The standard deviation (σ) is 5.74
Step-by-step explanation:
* Lets study the information to solve the problem
- The values of the data are 27 , 24 , 23 , 15 , 30 , 36 , 29 , 24
- They are eight values
* lets arrange them from small to big
∴ The values are 15 , 23 , 24 , 24 , 27 , 29 , 30 , 36
* Now lets solve the problem
# The range
- It is the difference between the largest and the smallest values
∵ The largest value is 36
∵ The smallest value is 15
∴ The range = 36 - 15 = 21
* The range is 21
# The interquartile range
- The steps to find the interquartile range is:
1- Arrange the values from the smallest to the largest
∴ The values are 15 , 23 , 24 , 24 , 27 , 29 , 30 , 36
2- Find the median
- The median is the middle value after arrange them
* If there are two values in the middle take their average
∵ The values are 8 then the 4th and the 5th are the values
∵ The 4th is 24 and the 5th is 27
∴ The median = [tex]\frac{24+27}{2}=\frac{51}{2}=25.5[/tex]
∴ The median is 25.5
3- Calculate the median of the lower quartile
- The lower quartile is the median of the first half data values
∵ There are 8 values
∴ The first half is the first four values
∴ The first half values are 15 , 23 , 24 , 24
∵ The middle values are 23 and 24
∴ The median of lower quartile = [tex]\frac{23+24}{2}=\frac{47}{2}= 23.5[/tex]
- Similar find the median of the upper quartile
- The upper quartile is the median of the second half data values
∵ There are 8 numbers
∴ The second half is the last four values
∴ The second half values are 27 , 29 , 30 , 36
∵ The middle values are 29 and 30
∴ The median of upper quartile = [tex]\frac{29+30}{2}=\frac{59}{2}=29.5[/tex]
4- The interquartile range (IQR) is the difference between the upper
and the lower medians
∴ The interquartile range = 29.5 - 23.5 = 6
* The interquartile range is 6
# The variance
- The variance is the measure of how much values in a set of data are
likely to differ from the mean value of the same data
- The steps to find the variance
1- Find the mean of the data
∵ The mean = sum of the data ÷ the number of the values
∵ The sum = 15 + 23 + 24 + 24 + 27 + 29 + 30 + 36 = 208
∵ The number of values is 8
∴ The mean = [tex]\frac{208}{8}=26[/tex]
∴ The mean is 26
2- Subtract the mean from each value and square the answer
∴ 15 - 26 = -11 ⇒ (-11)² = 121
∴ 23 - 26 = -3 ⇒ (-3)² = 9
∴ 24 - 26 = -2 ⇒ (-2)² = 4
∴ 24 - 26 = -2 ⇒ (-2)² = 4
∴ 27 - 26 = 1 ⇒ (1)² = 1
∴ 29 - 26 = 3 ⇒ (3)² = 9
∴ 30 - 26 = 4 ⇒ (4)² = 16
∴ 36 - 26 = 10 ⇒ (10)² = 100
3- Add all of these squared answer and divide the sum by the number
of the values
∴ The sum = 121 + 9 + 4 + 4 + 1 + 9 + 16 + 100 = 264
∵ They are 8 values
∴ The variance (σ²) = [tex]\frac{264}{8}=33[/tex]
* The variance is 33
# The standard deviation
- It is the square root of the variance
∵ The variance = 33
∴ The standard deviation (σ) = √33 = 5.74
* The standard deviation (σ) is 5.74
