Maths / Basic Statistics(1)
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Basic Statistics
Measures of Central Tendency
The mean is the statistical term for the average.
The mean is calculated by adding all scores then dividing by the number of scores. |
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Mean = |
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to add |
The median is the middle score or average of the two middle scores (once the scores are arranged in order).
The mode is the score with the highest frequency.
Example: Below are the wages of ten employees in a small business: $220 $230 $290 $275 $265 $250 $1500 $220 $220 $240
(a)Calculate the mean wage
(b)Calculate the median wage
(c)Calculate the mode wage
(d)Does the mean, median or mode give the best measure of a typical wage in the business?
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(a) |
Mean = |
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220+230+290+275+265+250+1500+220+220+240 |
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3710 |
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10 |
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10 |
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$371.00 |
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(b) |
Median (remember must be in order) |
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220 |
220 |
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240 |
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290 |
1500 |
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Median = |
240+250 |
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2 |
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=$245
(c)Mode = $220.00 (as it occurs 3 times)
(d)In this case, the median is the best measure of the typical score, because the mode is the lowest wage and the mean is inflated by the $1500.00
Practice Questions:
Calculate the mean, median, and mode of:
1.4, 8, 3, 5, 5
2.16, 24, 30, 35, 23, 11, 45, 28, 16, 16
3.9.2, 9.7, 8.8, 8.1, 5.6, 7.5, 8.5, 6.4, 9.2
Answers: |
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1. |
Mean |
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4+8+3+5+5 |
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25 |
5 |
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5 |
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5 |
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Median = |
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8 |
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Median = 5 |
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Mode |
= 5 |
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̅ |
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2. |
Mean |
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244 |
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10 |
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24.4 |
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Median = |
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16 |
23 |
24 |
28 |
30 |
35 |
45 |
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16 |
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16 |
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= 23.5 |
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Mode |
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16 |
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3. |
Mean |
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73 |
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9 |
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8.11 |
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Median = |
5.6 |
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7.5 |
8.1 |
8.5 |
8.8 |
9.2 |
9.2 |
9.7 |
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8.5 |
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Mode |
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9.2 |
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Measures of Spread |
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Range = highest score - lowest score |
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Lower Quartile |
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lowest1 |
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25% of the scores |
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Upper Quartile |
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highest3 |
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25% of the scores |
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Interquartile Range |
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(IQR) |
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upper3 quartile1 |
– lower quartile |
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middle 50% of the scores |
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Standard Deviation |
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a measure of how much a typical score in a data set differs |
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from the mean |
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σ = |
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2 |
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∑( −̅) |
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Example:
Here are the number of home runs scored in a series of base ball matches: 12 9 4 6 5 8 9 4 10 2
Calculate the
a)Range
b)Interquartile range
c)Standard deviation
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Range |
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Top score – bottom score |
=12 – 2
=10
(b)Write the data in ascending order.
2 |
4 |
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5 |
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10 |
12 |
Divide the data set in two |
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Lower quartile |
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the median of the lower half |
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4 |
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Upper quartile |
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the median of the upper half |
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9 |
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So IQR |
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9 – 4 |
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5 |
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(c) |
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2 |
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2- 6.9 = -4.9 |
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24.01 |
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4 |
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-2.9 |
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8.41 |
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4 |
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-2.9 |
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8.41 |
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5 |
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-1.9 |
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3.61 |
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6 |
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0.9 |
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0.81 |
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8 |
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1.1 |
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1.21 |
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9 |
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2.1 |
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4.41 |
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9 |
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2.1 |
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4.41 |
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10 |
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3.1 |
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9.61 |
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12 |
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5.1 |
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26.01 |
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= 69 |
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= 92.11 |
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̅= 6.9 |
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(−̅) |
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Standard deviation |
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σ = |
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2 |
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=9210.11
=3.03
Practice Questions:
For the following sets of data, calculate the:
(a)Range
(b)Interquartile range
(c)Standard deviation
1.3, 5, 9, 2, 7, 1, 6, 5
2.11, 8, 7, 12, 10, 11, 14
3.25, 15, 78, 35, 56, 41, 17, 24
Answers
1(a) Range = |
9 – 1 |
=8
(b) |
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3 |
5 |
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7 |
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Lower quartile= |
2.5 |
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Upper quartile = |
6.5 |
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IQR |
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6.5 -2.5 |
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=4
(c) |
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x |
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1 |
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-3.75 |
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14.06 |
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2 |
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-2.75 |
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7.56 |
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3 |
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-1.75 |
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3.06 |
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5 |
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0.25 |
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0.06 |
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5 |
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0.25 |
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0.06 |
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6 |
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1.25 |
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1.56 |
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7 |
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2.25 |
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5.06 |
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9 |
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4.25 |
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18.06 |
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= 38 |
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= 49.48 |
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Standard deviation |
σ = |
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498.48 |
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= 2.49 |
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2(a) Range = |
14 - 7 |
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(b) |
7 |
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11 |
11 |
12 |
14 |
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Median |
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11 |
(don’t include in IQR Calculation) |
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Bottom half |
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7 |
8 |
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Top half |
11 |
12 |
14 |
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Lower quartile= |
8 |
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Upper quartile = |
12 |
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IQR |
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12 - 8 |
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=4
(c)
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x |
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7 |
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-3.43 |
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11.76 |
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8 |
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-2.43 |
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5.9 |
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10 |
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-0.43 |
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0.18 |
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11 |
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0.57 |
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0.32 |
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11 |
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1.57 |
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2.46 |
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12 |
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2.57 |
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6.6 |
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14 |
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4.57 |
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20.88 |
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̅= 10.43 |
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= 48.1 |
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= 73 |
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Standard deviation |
σ = 487.1 |
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= 2.62 |
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3.(a) Range = |
78 -15 |
=63
(b) |
15 |
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24 |
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35 |
41 |
56 |
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Lower quartile= |
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Upper quartile = |
2 |
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17+24 |
41+56 |
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20.5 |
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48.5 |
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IQR |
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48.5 – 20.5 |
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=28
(c) |
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x |
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15 |
-21.4 |
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457.96 |
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17 |
-19.4 |
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376.36 |
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24 |
-12.4 |
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153.76 |
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25 |
-11.4 |
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129.96 |
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35 |
-1.4 |
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1.96 |
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41 |
4.6 |
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21.16 |
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56 |
19.6 |
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384.16 |
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78 |
41.6 |
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1730.56 |
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̅ |
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291 |
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=3255.88 |
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=36.4 |
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Standard deviation = σ
= 32558 .88
= 20.17
Frequency Histograms
There are 38 houses in a suburban street. The following table represents the number of people in each house:
No. of People |
Frequency |
1 |
1 |
2 |
4 |
3 |
10 |
4 |
15 |
5 |
8 |
Show this information in a histogram.
Frequency Histogram - suburban street
Frequency
16
14
12
10
8
6
4
2
0
1 |
2 |
3 |
4 |
5 |
Number of People in a House
Practise Question:
The marks out of 20 received by 30 students in a book review assignment are given below:
Mark |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
Frequency |
2 |
7 |
6 |
5 |
4 |
2 |
3 |
0 |
1 |
Draw a frequency histogram to represent this data.