19 lo4 measures of central tendency and spread
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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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Grade 12 Mathematical Literacy
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.
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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
50th percentile = )( 124100
50+
= 12,5= (25+30)2= 50th percentile = 27,5
75th percentile = )( 124100
75+
= 18,75= 75th percentile = 35,75
90th percentile = )( 124100
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
=
+
,
)(
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