19 LO4 Measures of Central Tendency and Spread

8/4/2019 19 LO4 Measures of Central Tendency and Spread

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Grade 12 Mathematical Literacy

LO4 Measures of central tendency and spread: Mean,mode, median, variance, deviation, percentiles,quartiles, range and outlier.

1. Calculate the mean, median and mode for each of the followingdatasets:

1.1 {43; 61; 18; 22; 46; 1 003; 35}

1.2 {14; 24; 35; 46; -46; -47; 55; -55; -15}

1.3 {121; 454; 76; 65; 92; 111; 76; 131; 990; 111}

1.4 How would the mean differ if the outlier(a data value significantly different from the other values in the set)were excluded from 1.1 above?

2. The variance of a set of data is the measure of the averagedifference between each number in the dataset and the mean (

n

)x-(xofSum 2=

2 ), while the standard deviation () is the

square root of the variance ( 2 ). Consider the following set ofdata and determine the mean, variance and standard deviation.

{5; 10; 15; 20; 25; 30; 35; 40}

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3. Consider the following dataset: {1; 5; 6; 8; 9; 13} The mean forthis dataset is 7.

3.1 Copy and complete the table below.

x 1 5 6 8 9 13xx 1-7 = -6 5-7 = -2

2)( xx (-6) = 36

3.2 Determine the variance of the dataset.

3.3 Determine the standard deviation of the dataset.

4. If the standard deviation of asset of numbers is 1, what does thisimply about the set?

5. Write down the lower, middle and upper quartiles of the followingdata and then the inter-quartile range.

{21; 22; 22; 23; 25; 26; 27}

6. Write down the 25th, 75th and 90th percentile of the following data:

21; 30; 33; 25; 20; 30; 35; 25; 20; 30; 35; 40;30; 37; 35; 25; 20; 40; 20; 25; 25; 30; 20; 20



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7. Mrs Smith, the Mathematical Literacy teacher, recorded her Grade12 examination results as follows.

7.1 Determine the mean.

7.2 Determine the mode.

7.3 Determine the median.

7.4 Determine the range.

7.5 Determine the variance.

7.6 Determine the standard deviation.

7.7 Determine the quartiles.

7.8 Determine the 60th percentile.


38 78 29 72 46

48 65 78 82 5967 47 90 58 69

39 39 66 48 80

88 48 69 48 92


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ANSWERS

1.1 {43; 61; 18; 22; 46; 1 003; 35}

Mean:7

3510034622186143 )( ++++++

= 175,43

Median: 18; 22; 35; 43; 46; 61; 1 003= 43

Mode: 18; 22; 35; 43; 46; 61; 1003They are all the mode as they appear an equal number oftimes.

1.2 {14; 24; 35; 46; -46; -47; 55; -55; -15}

Mean:9

155555474646352414 )( ++++

= 1,22

Median: -55; -47; -46; -15; 14; 24; 35; 46; 55= 14

Mode: 55; -47; -46; -15; 14; 24; 35; 46; 55

1.3 {121; 454; 76; 65; 92; 111; 76; 131; 990; 111}

Mean:10

11199013176111926576454121 )( +++++++++

Median: 65; 76; 76; 92; 111; 111; 121; 131; 454; 990

=2

111111 )( +

= 111

Mode: 76 and 111



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1.4 Mean:6

354622186143 )( +++++

= 37,5The mean is more realistic and more in line with the median.

2. {5; 10; 15; 20; 25; 30; 35; 40}

Mean:8

403530252015105 )( +++++++

= 22,5

Variance: 2

8

52235522305222552220522155221052252222222 ),(),(),(),(),(),(),( ++++++

= 82530625156255625625625562515625306 ,,,,,,,, +++++++

= 131,25

Standard deviation: 25131 ,=

= 11,46

3.1x 1 5 6 8 9 13

xx

1-7 = -6 5-7 = -2 6-7 = -1 8-7 = 1 9-7 = 2 13-7 = 62)( xx (-6) = 36 4 1 1 4 36

3.2 Variance:

6

364114362 +++++=

67132 ,=

3.3 Standard deviation: 6713,=

= 3,7

4. The numbers in the set of data are not widely dispersed, they areclose together. The closer the data are together, the nearer thestandard deviation is to 0.

5. {21; 22; 22; 23; 25; 26; 27}



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There are 7 data scores, so the median is 23.

21 22 22 23 25 26 27

Q1 Q2 Q3

Inter-quartile range: Q3 Q1 = 26 22 = 4



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6. First sort the data in numerical order:

20; 20; 20; 20; 20; 20; 21; 25; 25; 25; 25; 25; 30; 30; 30; 30; 33;35; 35; 35; 35; 35 ; 37; 40

25th percentile = )( 124100

25+

= 6,25= 25th percentile = 20,25


50+

= 12,5= (25+30)2= 50th percentile = 27,5


75+

= 18,75= 75th percentile = 35,75


90+

= 22,5

= 90

th

percentile = 36



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7.1 Mean:25

1543= 61,72

7.2 Mode: 48 (appears 3 times)

7.3 Median: 65

7.4 Range: 92 29 = 63

7.5 Variance:

Data Mean Difference ( xx ) Square 2)( xx

29 61.72 -32.72 1070.5984

38 61.72 -23.72 562.6384

39 61.72 -22.72 516.1984

39 61.72 -22.72 516.1984

46 61.72 -15.72 247.1184

47 61.72 -14.72 216.6784

48 61.72 -13.72 188.2384

48 61.72 -13.72 188.2384

48 61.72 -13.72 188.2384

48 61.72 -13.72 188.2384

58 61.72 -3.72 13.8384

59 61.72 -2.72 7.3984

65 61.72 3.28 10.758466 61.72 4.28 18.3184

67 61.72 5.28 27.8784

69 61.72 7.28 52.9984

69 61.72 7.28 52.9984

72 61.72 10.28 105.6784

78 61.72 16.28 265.0384

78 61.72 16.28 265.0384

80 61.72 18.28 334.1584

82 61.72 20.28 411.2784

88 61.72 26.28 690.6384

90 61.72 28.28 799.7584

92 61.72 30.28 916.8784

7855.04

Variance =25

0478552 ,=



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= 314,2

7.6 Standard Deviation: 2314 ,=

= 17,73

7.7 Quartiles:29 38 39 39 46 47 48 48 48 48 58 59 65 66 67 69 69 72 78 78 80 82 88 90 92

Q1 Q2 Q3

Q1 = 5472

4847,

)(=

+

Q2 = 65 Q3= 782

7878=

+ )(

7.8 60th percentile:

68

615

125100

60

=

+

,

)(


19 LO4 Measures of Central Tendency and Spread

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