PLAGIARISM FREE "A" WORK

I

RANGE, MODE, MEAN, MEDIAN, AND

STANDARD DEVIATION

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VIDEO RECORDING: https://youtu.be/jzpjp_2NIpA Problem: The following data set is: 121 110 135 147 121 152 160 166 162 125 141 159 164 140 155 Find the range, mode, mean, median, and standard deviation of the above data set. Solution: There are two types of data sets in statitics: population and sample. Population is a collection of outcomes, responses, mearsurements, or counts of interest. Sample is a subset or part of a population. The range, mean, median, and mode are calculated the same way for both types of data sets.

a. Find the range.

𝑅𝑎𝑛𝑔𝑒 = (𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑒𝑛𝑡𝑟𝑦) − (𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑑𝑎𝑡𝑎 𝑒𝑛𝑡𝑟𝑦) List the data in order.

110 121 121 125 135 140 141 147 152 155 159 160 162 164 166

𝑅𝑎𝑛𝑔𝑒 = 166 − 110

𝑅𝑎𝑛𝑔𝑒 = 56

b. Find the mode The mode of a data is the data entry that occurs most of the time. A data set can have more than one mode, one mode, or no mode at all.

110 121 121 125 135 140 141 147 152 155 159 160 162 164 166

The mode for this data is 121 since it appears twice.

c. Find the mean The mean of a data set is the sum of the data entries divided by the number of entries. It is the average of the data. The following are formulas how the mean is represented for population and sample means.

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑎𝑛 (𝜇) = Σ𝑥

𝑁 𝑜𝑟 𝑆𝑎𝑚𝑝𝑙𝑒 𝑀𝑒𝑎𝑛 (�̅�) =

Σ𝑥

𝑛

where 𝑥 = 𝑑𝑎𝑡𝑎 𝑒𝑛𝑡𝑟𝑦 𝑁 𝑜𝑟 𝑛 = 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑒𝑛𝑡𝑟𝑖𝑒𝑠 Therefore the mean for this data set is 110 + 121 + 121 + 125 + 135 + 140 + 141 + 147 + 152 + 155 + 159 + 160 + 162 + 164 + 166

15

𝑀𝑒𝑎𝑛 = 2,158

15 = 143.8666667

𝑀𝑒𝑎𝑛 ≈ 143.87

d. Find Median Median is the value in the data set that is in the middle. If there is an even number of data entries then the median is the mean of the two middle data entries. If there is an odd number of data entries then the median is the middle entry. Since the following data has an odd number of entries, 15, then the median is 147.

110 121 121 125 135 140 141 147 152 155 159 160 162 164 166 If we had the following even number of entries:

121 125 135 140 141 147 152 155 159 160 162 164 Then the median would be:

𝑀𝑒𝑑𝑖𝑎𝑛 = 147 + 152

2 =

299

2

𝑀𝑒𝑑𝑖𝑎𝑛 = 149.5

e. Find the standard deviation. Standard deviation measures the variation of the data set about the mean. The following are formulas how standard deviation is represented for population and sample deviations.

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = √ Σ(𝑥 − 𝜇)2

𝑁

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = √ Σ(𝑥 − �̅�)2

𝑛 − 1

𝑀𝑒𝑎𝑛 (𝜇 𝑜𝑟 �̅�) = 143.87

110 121 121 125 135 140 141 147 152 155 159 160 162 164 166

𝒙 (𝒙 − 𝝁) 𝒐𝒓 ( 𝒙 − �̅� ) (𝒙 − 𝝁)𝟐 𝒐𝒓 ( 𝒙 − �̅� )𝟐 110 -33.87 1147.1769 121 -22.87 523.0369 121 -22.87 523.0369 125 -18.87 356.0769 135 -8.87 78.6769 140 -3.87 14.9769 141 -2.87 8.2369 147 3.13 9.7969 152 8.13 66.0969 155 11.13 123.8769 159 15.13 228.9169 160 16.13 260.1769 162 18.13 328.6969 164 20.13 405.2169 166 22.13 489.7369

Sum = 4,563.7335

Population standard deviation:

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = √ Σ(𝑥 − 𝜇)2

𝑁

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = √ 4,563.7335

15

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = √304.2489

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) = 14.44273201

𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝜎) ≈ 14.44 Sample standard deviation:

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = √ Σ(𝑥 − �̅�)2

𝑛 − 1

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = √ 4,563.7335

15 − 1

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = √ 4,563.7335

14

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = √325.9809643

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) = 18.05494293

𝑆𝑎𝑚𝑝𝑙𝑒 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 (𝑠) ≈ 18.05