4. Centre and Spread

Lesson

When we find the median value of a data set, we are finding a central value with the property that there are as many data points below this value as there are above it. That is, we are finding a value that splits the data set into two equal parts.

We can extend this idea in order to define the quartiles. These are three numbers that split a data set into four equal parts–so the first quartile is the median of the lower half of the set, and the third quartile is the median of the upper half. (The second quartile is, of course, just the median of the whole set.)

Going another step further we could use $9$9 numbers to split the data into $10$10 equal parts called deciles.

A step further from this we could split the data into $100$100 equal parts called percentiles.

Quartiles, percentiles and deciles are useful to gauge the rank of a score. For example, as the deciles split the data into ten equal parts the first decile $D_1$`D`1, will cut off the bottom $10%$10% of scores, the $4$4th decile $D_4$`D`4 will cut off the bottom $40%$40% of scores and the $9$9th decile $D_9$`D`9 will cut off the bottom $90%$90% of scores–leaving $10%$10% of scores at or above it.

Similarly, the $15$15th percentile $P_{15}$`P`15, will cut off the bottom $15%$15% of scores, the $60$60th percentile $P_{60}$`P`60 will cut off the bottom $60%$60% of scores and the $95$95th percentile $P_{95}$`P`95 will cut off the bottom $95%$95% of scores–leaving $5%$5% of scores at or above it.

Height and weight charts, as well as university entrance rankings, are often given using deciles or percentiles. If you found you were in the $85$85th percentile for height, this would mean you are in the tallest $15%$15% of your age group.

Clearly, the median should be the same as the $50$50th percentile, the first quartile should be the same as the $25$25th percentile and the third quartile should be the same as the $75$75th percentile. Similarly, for example, the $4$4th decile is the same as the $40$40th percentile.

For the following set of twenty numbers:

$12$12 | $15$15 | $16$16 | $16$16 | $18$18 | $19$19 | $19$19 | $22$22 | $24$24 | $25$25 | $25$25 | $25$25 | $26$26 | $27$27 | $29$29 | $33$33 | $35$35 | $38$38 | $42$42 | $44$44 |

**(a)** Find the value of the $2$2nd decile.

**Think:** Make sure the data is ordered and then split the data into $10$10 equal parts. There are $20$20 scores, so we want $2$2 scores in each part.

**Do:**

The second decile $D_2$`D`2 falls between the scores $16$16 and $18$18, finding the average of these two scores we get:

$D_2$D2 |
$=$= | $\frac{16+18}{2}$16+182 |
Finding the average of $16$16 and $18$18 |

$=$= | $17$17 |
Simplifying the fraction |

Hence, the second decile is $17$17. That is $20%$20% of scores fall below this value and $80%$80% are above.

**(b)** What percentage of scores fall below the $8$8th decile?

**Think:** The $8$8th decile is the same as the $80$80th percentile.

**Do:** $80%$80% of scores fall below the $8$8th decile.

**(c)** Find the $90$90th percentile.

**Think:** The $90$90th percentile separates the bottom $90%$90% from the top $10%$10%, this is the same as the $9$9th decile.

**Do:** The $90$90th percentile is equal to $D_9$`D`9, this falls between the values $38$38 and $42$42, hence:

$P_{90}$P90 |
$=$= | $\frac{38+42}{2}$38+422 |
Finding the average of $38$38 and $42$42 |

$=$= | $40$40 |
Simplifying the fraction |

**(d)** Find the median.

**Think:** The median is the same as the $50$50th percentile or the $5$5th decile.

**Do:** The median is between the values $25$25 and $25$25. Hence, the median is $25$25.

**(e)** What value separates the bottom $70%$70% of scores from the top $30%$30% of scores?

**Think:** We want the value of $D_7$`D`7.

**Do:** $D_7$`D`7 falls between the values $27$27 and $29$29.

$D_7$D7 |
$=$= | $\frac{27+29}{2}$27+292 |
Finding the average of $27$27 and $29$29 |

$=$= | $28$28 |
Simplifying the fraction |

The following is based on population data for the weights of $9.5$9.5 month old female infants.

- The median is $8.7$8.7 kg
- The $3$3rd percentile is $7.0$7.0 kg
- The $25$25th percentile is $8.1$8.1 kg
- The $9$9th decile is $10.0$10.0 kg
- The $97$97th percentile is $10.7$10.7 kg

**(a)** Is a female infant weighing $10$10 kg large for this age? What percentage of $9.5$9.5 month old female infants will weigh more than this?

**Think:** $10$10 kg is the $9$9th decile, what percentile is this?

**Do:** The $9$9th decile is the $90$90th percentile. This means $90%$90% of girls will weigh less than this. Hence, yes this is large for this age group and only $10%$10% of girls will weigh equal to or more than this.

**(b)** If $3%$3% of girls are lighter than Molly, how much does Molly weigh?

**Think:** Molly's weight is at the third percentile.

**Do:** Molly weighs $7.0$7.0 kg.

**(c)** What weight separates the lightest quarter of female infants of this age?

**Think:** The value separating the lightest quarter of infants will be the first quartile–this is the same as the $25$25th percentile.

**Do:** The $25$25th percentile is $8.1$8.1 kg.

There are several different methods for determining the percentiles of a data set, each giving slightly different results. Some methods, like our method above, average the two closest values when a percentile falls between values, others may round up to the nearest score. The differences in these calculations are not significant when the data sets are large.

Consider the following data set containing $20$20 scores.

$11$11 | $12$12 | $18$18 | $18$18 | $21$21 | $22$22 | $22$22 | $25$25 | $27$27 | $28$28 | $29$29 | $29$29 | $29$29 | $30$30 | $32$32 | $36$36 | $40$40 | $41$41 | $45$45 | $47$47 |

Find the value of the second decile.

What percentage of scores fall below the seventh decile?

Find the $80$80th percentile.

Find the median.

Find the decile that separates the bottom $40%$40% of scores from the top $60%$60% of scores.

The attached chart shows the range of heights for boys aged $2$2 to $18$18 years.

(Credit: State Government of Victoria, Department of Education and Training)

Lachlan is $15$15 and his height is at the $9$9th decile. How tall could he be?

$177-181$177−181 cm

A$167-172$167−172 cm

B$158-161$158−161 cm

C$177-181$177−181 cm

A$167-172$167−172 cm

B$158-161$158−161 cm

C

The following information is based on population data for the heights of fifteen year old males.

- The third percentile is $154.6$154.6 cm.
- The $25$25th percentile is $164.8$164.8 cm.
- The fifth decile is $170.1$170.1 cm.
- The $95$95th percentile is $182.4$182.4 cm.

If Xavier is $170$170 cm tall, approximately what percentage of boys in this age group would be taller?

Approximately what percentage of fifteen year old males are between $154.6$154.6 cm and $170.1$170.1 cm?

If $40$40 fifteen year old males are randomly selected, how many would you expect to be taller than $182.4$182.4 cm?

calculate quartiles from a dataset [complex]

interpret quartiles, deciles and percentiles from a graph [complex]