# Mean, median, mode, variance & standard deviation

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## Transcript Of Mean, median, mode, variance & standard deviation

Subject: Statistics Created by: Marija Stanojcic
Revised: 10/9/2018
Mean, median, mode, variance & standard deviation

MEAN β average value of the data set

symbol(s): π β ππππ’πππ‘πππ ππππ

π₯Μ β sample mean,

π₯Μ = π₯1 + π₯2 + ...+π₯π = βππ = 1 π₯π

π

π

Example 1: Calculate the mean for the data set: 1, 4, 4, 6, 10.

π₯Μ = β5π = 1 π₯π = π₯1 + π₯2+π₯3+π₯4+π₯5 = 1 + 4 + 4 + 6 + 10 = 25 = 5

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5

5

5

MEDIAN β βthe middle numberβ, a number that splits data set in half How to find median? Step 1: Arrange data in increasing order Step 2: Determine how many numbers are in the data set = n Step 3: If n is odd: Median is the middle number If n is even: Median is the average of two middle numbers
Example 2: Find the median for the data set: 34, 22, 15, 25, 10. Step 1: Arrange data in increasing order 10, 15, 22, 25, 34 Step 2: There are 5 numbers in the data set, n = 5. Step 3: n = 5, so n is an odd number Median = middle number, median is 22.
Example 3: Find the median for the data set: 19, 34, 22, 15, 25, 10. Step 1: Arrange data in increasing order 10, 15, 19, 22, 25, 34 Step 2: There are 6 numbers in the data set, n = 6. Step 3: n = 6, so n is an even number Median = average of two middle numbers median = 19 + 22 = 20.5
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Notes: Mean and median donβt have to be numbers from the data set! Mean and median can only take one value each. Mean is influenced by extreme values, while median is resistant.

Subject: Statistics Created by: Marija Stanojcic
Revised: 10/9/2018
Mean, median, mode, variance & standard deviation
MODE β The most frequent number in the data set Example 4: Find the mode for the data set: 19, 19, 34, 3, 10, 22, 10, 15, 25, 10, 6.
The number that occurs the most is number 10, mode = 10. Example 5: Find the mode for the data set: 19, 19, 34, 3, 10, 22, 10, 15, 25, 10, 6, 19.
Number 10 occurs 3 times, but also number 19 occurs 3 times, since there is no number that occur 4 times both numbers 10 and 19 are mode, mode = {10, 19}. Notes: Mode is always the number from the data set.
Mode can take zero, one, or more than one values. (There can be zero modes, one mode, two modes, β¦)
Shape of a Data Set
Relationship between mean and median, in most cases, can describe the shape of data set (histogram).

(a) Median < Mean skewed to the RIGHT positively skewed

(b) Median = Mean symmetric

(c) Median > Mean skewed to the LEFT negatively skewed

Subject: Statistics Created by: Marija Stanojcic
Revised: 10/9/2018
Mean, median, mode, variance & standard deviation

The difference between value π₯ and population mean π , π₯ β π is deviation.

VARIANCE measures how far the values of the data set are from the mean, on average. The average of the squared deviations is the population variance.

Symbol(s): Ο2 β population variance s2 β sample variance

Ο2 = βππ=1(π₯π β π₯Μ)2
π
s2 = βππ=1(π₯π β π₯Μ)2
πβ1

STANDARD DEVIATION β square root of the variance

Symbol(s): Ο β population standard deviation

Ο = βΟ2 = ββππ=1(π₯π β π₯Μ)2
π

s β sample standard deviation

s = βs2 = ββππ=1(π₯π β π₯Μ)2
πβ1

How to compute variance and standard deviation? (of a sample) For variance do steps 1-5, for standard deviation do steps 1-6.
Step 1 β Compute the sample mean π₯Μ Step 2 β Calculate the difference of π₯π β π₯Μ , for each value in the data set Step 3 β Calculate the squared difference (π₯π β π₯Μ)2, for each value in the data set Step 4 β Sum the squared differences βππ=1(π₯π β π₯Μ)2 Step 5 β Divide the sum of squared differences with n-1, π£πππππππ = π 2 = βππ=1(π₯π β π₯Μ)2
πβ1
Step 6 β ONLY FOR STANDARD DEVIATION Calculate the squared root of variance, π π‘ππππππ πππ£πππ‘πππ = βπ£πππππππ
s = ββππ=1(π₯π β π₯Μ)2
πβ1

References: Essential Statistics, 2nd Ed., Navidi, Monk
DataModeDeviationVarianceStatistics