Question

find the range, interquartile range, and mean absolute deviation for the following data set: 58, 60, 62, 62, 65, 68, 70

Explanation:

Step1: Calculate the range

The range is the difference between the maximum and minimum values.
$70 - 58=12$

Step2: Find the median (Q2)

The data - set has 7 values. The median is the 4th value when the data is ordered. So, $Q2 = 62$.

Step3: Find Q1 and Q3

The lower half of the data is $58,60,62$. The median of the lower half ($Q1$) is 60. The upper half of the data is $65,68,70$. The median of the upper half ($Q3$) is 68.

Step4: Calculate the inter - quartile range (IQR)

$IQR=Q3 - Q1=68 - 60 = 8$

Step5: Calculate the mean

The mean $\bar{x}=\frac{58 + 60+62+62+65+68+70}{7}=\frac{445}{7}\approx63.57$

Step6: Calculate the absolute deviations

$|58 - 63.57|=5.57$, $|60 - 63.57| = 3.57$, $|62 - 63.57|=1.57$, $|62 - 63.57|=1.57$, $|65 - 63.57| = 1.43$, $|68 - 63.57|=4.43$, $|70 - 63.57|=6.43$

Step7: Calculate the mean absolute deviation (MAD)

$MAD=\frac{5.57+3.57+1.57+1.57+1.43+4.43+6.43}{7}=\frac{24.5}{7}=3.5$

Answer:

Range: 12, Inter - quartile range: 8, Mean absolute deviation: 3.5