The number of patients treated each day by a dentist during a 20 -day period is shown in the following stem-and-leaf diagram. Describe a situation in which it would be to the dentist's advantage to use a mode as the average.

Given that, the stem and leaf plot is So the given data is 4 , 4 , 4 , 5 , 6 , 6 , 6 , 7 , 14 , 15 , 15 , 16 , 16 , 17 , 17 , 18 , 18 , 19 , 20 , 21 So, there are 20 observations. We know that, median of even number of observations is the mean of two middle observations. ⇒ Median = 15 + 15 2 = 15 Thus the median of the given stem and leaf plot is 15 . Here the mode of the given data set is 4 and 6 . Thus the dentists advantage to use a mode as the average is for example, being unable to pay a bill because of low earnings.

t 7 8 9 10 11 12 13 f 4 7 9 14 16 41 9 What feature of the data suggests that t ¯ is less than the median? Confirm whether or not this is the case

Given data is t 7 8 9 10 11 12 13 f 4 7 9 14 16 41 9 Data values along with cumulative frequency: t 7 8 9 10 11 12 13 f 4 7 9 14 16 41 9 Cumulative frequency 4 11 20 34 50 91 100 Since data appear in an ordered frequency table of individual values, we can use cumulative frequencies to investigate the positions of the values, knowing that the median = n + 1 2 th data value, where n is the number of data values in the set. Here, number of data values is n = ∑ f = 100 ⇒ Median = 100 + 1 2 th = 50 . 5 So we locate the median mid-way between 50 and 51 data values. So the median is the mean of 11 and 12 ⇒ median = 11 + 12 2 = 11 . 5 Hence, the median of the given data is 11 . 5 . Now, mean of the given data is t ¯ = Σ f t Σ f = 7 × 4 + 8 × 7 + 9 × 9 + 10 × 14 + 11 × 16 + 12 × 41 + 13 × 9 100 = 1090 100 = 10 . 9 Thus the value of t ¯ is less than median.

The following graph illustrates the times taken by 112 people to complete a puzzle. The median is used to divide these people into two groups. Find the median time taken by each of the groups.

Given: Cumulative frequency curves enable us to find the median and quartiles from a set of grouped data. To find the median, we draw a horizontal line across from the frequency axis to the curve at the n 2 th value and then down to the x -axis to determine the median data point. Here, n = 112 So, n 2 th value is 56 th value. Draw a line across from 56 on the vertical axis( y -axis) until it hits the curve and then draw a line vertically down until it touches the horizontal axis( x -axis). This is the median Q 2 . Draw a line across from 28 and 84 on the vertical axis( y -axis) until it hits the curve and then draw a line vertically down until it touches the horizontal axis( x -axis). This is the median Q 1 and Q 3 respectively. Thus, lower group median 28 t h approx 2 . 8 minutes and upper group median 84 t h approx 6 . 4 minutes.

EASY

AS and A Level

IMPORTANT

Earn 100

The number of patients treated each day by a dentist during a $20$ -day period is shown in the following stem-and-leaf diagram.

On eight of these $20$ days, the dentist arrived late to collect their son from school. If they decide to use their average number of patients as a reason for arriving late, would they use the median or the mean? Explain your answer.

Important Questions on Measures of Central Tendency

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 1

The number of patients treated each day by a dentist during a $20$ -day period is shown in the following stem-and-leaf diagram.

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Describe a situation in which it would be to the dentist's advantage to use a mode as the average.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 2

$t$	$7$	$8$	$9$	$10$	$11$	$12$	$13$
$f$	$4$	$7$	$9$	$14$	$16$	$41$	$9$

Find the median for the values of $t$ given in the following table.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 2

$t$	$7$	$8$	$9$	$10$	$11$	$12$	$13$
$f$	$4$	$7$	$9$	$14$	$16$	$41$	$9$

What feature of the data suggests that $\bar{t}$ is less than the median? Confirm whether or not this is the case

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 3

$t$	$4$	$5$	$6$	$7$	$8$
$f$	$14$	$13$	$4$	$12$	$15$

Find the median and the mode for the values of $t$ given in the following table.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 3

$t$	$4$	$5$	$6$	$7$	$8$
$f$	$14$	$13$	$4$	$12$	$15$

Find the median and the mode for the values of $t$ given in the following table. Give one positive and one negative aspect of using each of the median and the mode as the average value for $x .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 3

$t$	$4$	$5$	$6$	$7$	$8$
$f$	$14$	$13$	$4$	$12$	$15$

Some values in the table have been incorrectly recorded as $8$ instead of $4 .$ Find the number of incorrectly recorded values, given that the true median of $x$ is $5.5 .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>EXERCISE 2E>Q 4

The following graph illustrates the times taken by $112$ people to complete a puzzle.

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