2.6 Descriptive statistics: measuring the location of the data

Order the data from smallest to largest.

• 114,950
• 158,000
• 230,500
• 387,000
• 389,950
• 479,000
• 488,800
• 529,000
• 575,000
• 639,000
• 659,000
• 1,095,000
• 5,500,000

$M=\mathrm{488,800}$

$Q_1=\frac{230500+387000}{2}=308750$

$Q_3=\frac{639000+659000}{2}=649000$

$\mathrm{IQR}=649000-308750=340250$

$\left(1.5\right)\left(\mathrm{IQR}\right)=\left(1.5\right)\left(340250\right)=510375$

$Q_1-\left(1.5\right)\left(\mathrm{IQR}\right)=308750-510375=-201625$

$Q_3+\left(1.5\right)\left(\mathrm{IQR}\right)=649000+510375=1159375$

No house price is less than -201625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier .

For the IQRs, see the answer to the test scores example . The first data set has the larger IQR, so the scores between $\mathrm{Q3}$ and $\mathrm{Q1}$ (middle 50%) for the first data set are more spread out and not clustered about the median.

First data set

• $\left(\frac{3}{2}\right) \cdot \left(\mathrm{IQR}\right) = \left(\frac{3}{2}\right) \cdot \left(26.5\right) = 39.75$
• $\mathrm{Xmax} - \mathrm{Q3} = 99 - 82.5 = 16.5$
• $\mathrm{Q1} - \mathrm{Xmin} = 56 - 32 = 24$

Second data set

• $\left(\frac{3}{2}\right)\cdot \left(\mathrm{IQR}\right)=\left(\frac{3}{2}\right)\cdot \left(11\right)=16.5$
• $\mathrm{Xmax}-\mathrm{Q3}=98-89=9$
• $\mathrm{Q1}-\mathrm{Xmin}=78-25.5=52.5$

To find the percentiles, create a frequency, relative frequency, and cumulative relative frequency chart (see "Frequency" from the Sampling and Data Chapter ). Get the percentiles from that chart.

First data set

• $\text{30th \%ile (between the 6th and 7th values)} = \frac{(56 + 59)}{2} = 57.5$
• $\text{80th \%ile (between the 16th and 17th values)} = \frac{(84 + 84.5)}{2} = 84.25$

Second data set

• $\text{30th \%ile (7th value)}=78$
• $\text{80th \%ile (18th value)}=90$

30% of the data falls below the 30th %ile, and 20% falls above the 80th %ile.