Ronald Hui Tak Sun Secondary School HKDSE Mathematics Ronald Hui Tak Sun Secondary School

Homework SHW6-C1: Sam L SHW7-B1: Sam L SHW7-P1: Sam L SHW8-A1: Sam L SHW8-P1: Kelvin RE8: Sam L Ronald HUI

Standard Quiz Result of Standard Quiz Major Problems Full mark: 30 Highest: 8 (12) Lowest: 0 (x5) – Oh no!!! 2 more!!! Average: 4.17 (2.1) Number of students (10-14): 0 Number of students (0 – 9): 23 (Oh no!!! x2) Major Problems Did not start to read the questions Just give up!

Standard Quiz Result of Standard Quiz 5th: HUI Ka Ho, Ken (1) 1st: CHAN Chun Hang, Jason (3) HO Yat Wan, Owen (NEW) LI Chi Hin, Ronald (NEW) WONG Yin Man Marco (=) Congratulations!!! Well done!!!

Standard Quiz (24) Result of Standard Quiz Major Problems Full mark: 24 Highest: 17 (5) Lowest: 0 ( 3) Average: 7.43 (3.12) Number of students (10-11): 0 Number of students (0 – 9): 16 Major Problems Cannot read the questions carefully Use wrong formula

Standard Quiz (24) Result of Standard Quiz (24) 5th: WONG Vincent Wai Shun (2) TSE Pak Ho, Pako (NEW) YAN Tin Lok (=) 3rd: HO Yat Wan, Owen (NEW) WAN Hang Nam, Hanki (1) 2nd: WONG Yin Man Marco (8) 1st: CHAN Chun Hang, Jason (6) Congratulations!!!

Effects of Data Change on Measures of Dispersion Title page: Font size 36, bold, theme color of the chapter (red for geometry, blue for algebra, green for statistics)

Does the dispersion of the data change after adding 2 to each datum? Adding a Common Constant to Each Datum Data set A: 1, 2, 2, 3, 3, 3, 3, 4 Does the dispersion of the data change after adding 2 to each datum? 1 2 3 4 5 6 1 7 8 9 Data set A Data set B: 3, 4, 4, 5, 5, 5, 5, 6 Add 2 to each datum After adding 2 to each datum, the shape of the distribution remains unchanged. Data set B 1 2 3 4 5 6 1 7 8 9

Adding a Common Constant to Each Datum Data set A: 1, 2, 2, 3, 3, 3, 3, 4 Q1 Q2 Q3 Range = 4 – 1 = 3 1 2 3 4 5 6 1 7 8 9 Data set A Inter-quartile range = 3 – 2 = 1 Standard deviation = 0.857 Data set B: 3, 4, 4, 5, 5, 5, 5, 6 Add 2 to each datum Q1 Q2 Q3 Range = 6 – 3 = 3 The range remains unchanged. Data set B Inter-quartile range = 5 – 4 = 1 The inter-quartile range remains unchanged. Standard deviation = 0.857 1 2 3 4 5 6 1 7 8 9 The standard deviation remains unchanged.

In general, if a common constant is added to each datum of a data set, then: New range = original range New inter-quartile range = original inter-quartile range New standard deviation = original standard deviation If a common constant is subtracted from each datum of a data set, the measures of dispersion also remain unchanged.

Multiplying Each Datum by a Common Constant Data set A: 1, 2, 2, 3, 3, 3, 3, 4 Does the dispersion of the data change after multiplying each datum by 2? 1 2 3 4 5 6 1 7 8 9 Data set A Data set C: 2, 4, 4, 6, 6, 6, 6, 8 Multiply each datum by 2 After multiplying each datum by 2, the shape of the distribution is changed. The spread is greater. Data set C 1 2 3 4 5 6 1 7 8 9

Multiplying Each Datum by a Common Constant Data set A: 1, 2, 2, 3, 3, 3, 3, 4 Q1 Q2 Q3 Range = 4 – 1 = 3 1 2 3 4 5 6 1 7 8 9 Data set A Inter-quartile range = 3 – 2 = 1 Standard deviation = 0.857 Data set C: 2, 4, 4, 6, 6, 6, 6, 8 Multiply each datum by 2 Q1 Q2 Q3 Range = 8 – 2 = 6 The range of C is twice that of A. Data set C Inter-quartile range = 6 – 4 = 2 The inter-quartile range of C is twice that of A. Standard deviation = 1.71 1 2 3 4 5 6 1 7 8 9 The standard deviation of C is twice that of A.

In general, if each datum of a data set is multiplied by a positive common constant k, then: New range = original range  k New inter-quartile range = original inter-quartile range  k New standard deviation = original standard deviation  k Note: Since the measures of dispersion should be non-negative, if k is negative, then: New range = original range  (k) New inter-quartile range = original inter-quartile range  (k) New standard deviation = original standard deviation  (k)

Follow-up question The costs of 7 second-hand smartphones in a shop are as follows: $1300, $1700, $1800, $2000, $2700, $3900, $4800 Find the range, the inter-quartile range and the standard deviation of the costs of the smartphones. The shop owner decides to mark the price higher than the cost of these smartphones, find the range, the inter-quartile range and the standard deviation of the marked prices of these smartphones. (i) $100, (ii) 10%

Follow-up question The costs of 7 second-hand smartphones in a shop are as follows: $1300, $1700, $1800, $2000, $2700, $3900, $4800 Find the range, the inter-quartile range and the standard deviation of the costs of the smartphones. (a) Range = $4800  $1300 = $3500 Inter-quartile range = $3900  $1700 = $2200 Standard deviation = $1200 (cor. to 3 sig. fig.)

Follow-up question The shop owner decides to mark the price higher than the cost of these smartphones, find the range, the inter-quartile range and the standard deviation of the marked prices of these smartphones. (i) $100, (ii) 10% (b) (i) The range the marked prices = $3500 The inter-quartile range the marked prices = $2200 The standard deviation the marked prices = $1200 (cor. to 3 sig. fig.)

Follow-up question The shop owner decides to mark the price higher than the cost of these smartphones, find the range, the inter-quartile range and the standard deviation of the marked prices of these smartphones. (i) $100, (ii) 10% (b) (ii) The range of the marked prices = $[3500  (1 + 10%)] = $3850 The inter-quartile range of the marked prices = $[2200  (1 + 10%)] = $2420 The standard deviation of the marked prices  $[1197.617  (1 + 10%)] = $1320 (cor. to 3 sig. fig.)

Removing a Datum from a Set of Data Effect on range When a datum is removed from a set of data, the range will decrease only if the deleted datum is the largest datum or the smallest datum, and the removed datum is unique. Otherwise, the range will not be affected.

Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. There are 2 smallest data ‘12’ in the data set, i.e. the smallest datum is not unique. Range = 22 – 12 = 10 Case 1: Remove the largest datum, i.e. ‘22’ which is unique in the data set. New data set 12, 12, 14, 16, 20 New range 20 – 12 = 8 The range decreases.  The largest datum decreases to 20.

Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Range = 22 – 12 = 10 Case 2: Remove the smallest datum, i.e. ‘12’ which is not unique in the data set. New data set 12, 14, 16, 20, 22 New range 22 – 12 = 10 The range remains unchanged.  The smallest datum is still 12.

Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Range = 22 – 12 = 10 Case 3: Remove a datum other than the largest datum or the smallest datum, say ‘14’. New data set 12, 12, 16, 20, 22 New range 22 – 12 = 10 The range remains unchanged.

Effect on inter-quartile range Removing a datum from a set of data may change its upper quartile and/or the lower quartile. Therefore, the inter-quartile range may increase, decrease or remain unchanged.

New inter-quartile range Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Q1 Q3 Inter-quartile range = 20 – 12 = 8 Case 1: Remove a datum ‘14’. New data set 12, 12, 16, 20, 22 New inter-quartile range 21 – 12 = 9 The inter-quartile range increases.  Q1 remains unchanged but Q3 increases.

New inter-quartile range Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Q1 Q3 Inter-quartile range = 20 – 12 = 8 Case 2: Remove a datum ‘20’. New data set 12, 12, 14, 16, 22 New inter-quartile range 19 – 12 = 7 The inter-quartile range decreases.  Q1 remains unchanged but Q3 decreases.

New inter-quartile range Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Consider the data set: 12, 12, 14, 16, 20, 22. Q1 Q3 Inter-quartile range = 20 – 12 = 8 Case 3: Remove a datum ‘12’. New data set 12, 14, 16, 20, 22 New inter-quartile range 21 – 13 = 8 The inter-quartile range remains unchanged.  Both Q1 and Q3 increase.

Effect on standard deviation Since the standard deviation takes all data into account, it may increase, decrease or remain unchanged if a datum is removed.

New standard deviation New standard deviation Consider the data set: 12, 12, 14, 16, 20, 22. Mean = 16, standard deviation = 3.83 (cor. to 3 sig. fig.) Case 1: Remove a datum equal to the mean, i.e. ‘16’. New data set 12, 12, 14, 20, 22 New standard deviation 4.1952… The standard deviation increases. Case 2: Remove a datum further away from the mean, say ‘22’. New data set 12, 12, 14, 16, 20 New standard deviation 2.9933… The standard deviation decreases.

In general, 1. if a datum equal to the mean is removed from a data set which are not entirely the same, the distribution will become less concentrated about the mean. Therefore, the standard deviation will increase. 2. if a datum removed from a data set is further away from the mean than all other data, the distribution will become more concentrated about the mean. Therefore, the standard deviation will decrease.

Follow-up question Consider the following set of data. {3, 6, 8, 9, 12, 14} (a) Find the range and the standard deviation of the data. (b) If the datum ’14’ is removed, (i) find the range and the standard deviation of the remaining 5 data. (ii) will the range and the standard deviation increase, decrease or remain unchanged? State the reasons for the results obtained. (Give your answers correct to 3 significant figures if necessary.) (a) Range = 14 – 3 = 11 Standard deviation ≈ 3.6362 = 3.64 (cor. to 3 sig. fig.)

Follow-up question Consider the following set of data. {3, 6, 8, 9, 12, 14} (a) Find the range and the standard deviation of the data. (b) If the datum ’14’ is removed, (i) find the range and the standard deviation of the remaining 5 data. (ii) will the range and the standard deviation increase, decrease or remain unchanged? State the reasons for the results obtained. (Give your answers correct to 3 significant figures if necessary.) (b) (i) New range = 12  3 = 9 New standard deviation = 3.01 (cor. to 3 sig. fig.)

Follow-up question (b) (ii) Since the maximum datum is decreased from 14 to 12, Since the removed datum is further away from the mean than other data, removing it will make the distribution of data more concentrated about the mean. Therefore, the standard deviation is decreased. the range is decreased.

Adding a Datum to a Set of Data Effect on range When a datum is added to a set of data, the range will increase only if the added datum is larger than the largest datum or smaller than the smallest datum. Otherwise, the range will not be affected.

Case 1: Add a datum smaller than the smallest datum, say ‘10’. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Range = 26 – 12 = 14 Case 1: Add a datum smaller than the smallest datum, say ‘10’. New data set 10, 12, 14, 16, 18, 20, 20, 26 New range 26 – 10 = 16  The smallest datum decreases to 10. The range increases.

Consider the data set: 12, 14, 16, 18, 20, 20, 26. Range = 26 – 12 = 14 Case 2: Add a datum between the smallest and the largest data, say ‘13’. New data set 12, 13, 14, 16, 18, 20, 20, 26 New range 26 – 12 = 14  The smallest and the largest datum remain unchanged. The range remains unchanged.

Case 3: Add a datum greater than the largest datum, say ‘28’. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Range = 26 – 12 = 14 Case 3: Add a datum greater than the largest datum, say ‘28’. New data set 12, 14, 16, 18, 20, 20, 26, 28 New range 28 – 12 = 16  The largest datum increases to 28. The range increases.

Effect on inter-quartile range Adding a datum to a set of data may change its upper quartile and/or the lower quartile. Therefore, the inter-quartile range may increase, decrease or remain unchanged.

New inter-quartile range Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Q1 Q3 Inter-quartile range = 20 – 14 = 6 Case 1: Add a datum ‘19’. New data set 12, 14, 16, 18, 19, 20, 20, 26 New inter-quartile range 20 – 15 = 5 The inter-quartile range decreases.  Q1 increases, but Q3 remains unchanged.

New inter-quartile range Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Q1 Q3 Inter-quartile range = 20 – 14 = 6 Case 2: Add a datum ‘14’. New data set 12, 14, 14, 16, 18, 20, 20, 26 New inter-quartile range 20 – 14 = 6  Both Q1 and Q3 remain unchanged. The inter-quartile range remains unchanged.

New inter-quartile range Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Consider the data set: 12, 14, 16, 18, 20, 20, 26. Q1 Q3 Inter-quartile range = 20 – 14 = 6 Case 3: Add a datum ‘10’. New data set 10, 12, 14, 16, 18, 20, 20, 26 New inter-quartile range 20 – 13 = 7 The inter-quartile range increases.  Q1 decreases, but Q3 remains unchanged.

Effect on standard deviation Since the standard deviation takes all data into account, it may increase, decrease or remain unchanged if a datum is added.

New standard deviation New standard deviation Consider the data set: 12, 14, 16, 18, 20, 20, 26 Mean = 18, standard deviation = 4.28 (cor. to 3 sig. fig.) Case 1: Add a datum equal to the mean, i.e. ‘18’. New data set 12, 14, 16, 18, 18, 20, 20, 26 New standard deviation 4 The standard deviation decreases. Case 2: Add a datum further away from the mean, say ‘12’. New data set 12, 12, 14, 16, 18, 20, 20, 26 New standard deviation 4.4651… The standard deviation increases.

In general, 1. if a datum equal to the mean is added to a data set which are not entirely the same, the distribution will become more concentrated about the mean. Therefore, the standard deviation will decrease. 2. if a datum added to a data set is further away from the mean than all other data, the distribution will become less concentrated about the mean. Therefore, the standard deviation will increase.

Follow-up question The capacities (in mL) of the 6 cups in Andrew’s house are shown below. 90, 180, 250, 350, 500, 600 (a) Find the inter-quartile range and the standard deviation of the capacities of the cups. (b) If Andrew buys a new cup with capacity 430 mL, will the inter-quartile range and the standard deviation of the capacities of the cups increase, decrease or remain unchanged? If there is an increase/ decrease, find the corresponding changes. (Give your answers correct to 3 significant figures if necessary.)

Follow-up question The capacities (in mL) of the 6 cups in Andrew’s house are shown below. 90, 180, 250, 350, 500, 600 (a) Find the inter-quartile range and the standard deviation of the capacities of the cups. (a) Inter-quartile range = (500 – 180) mL = 320 mL Standard deviation ≈ 177.3336 mL = 177 mL (cor. to 3 sig. fig.)

Follow-up question (b) The capacities (in mL) of the 7 cups are: 90, 180, 250, 350, 430, 500, 600 Inter-quartile range = (500 – 180) mL = 320 mL Standard deviation ≈ 167.9893 mL ∴ Inter-quartile range remains unchanged. Decrease in the standard deviation ≈ (177.3336 167.9893) mL = 9.34 mL (cor. to 3 sig. fig.)