# Mean, median, mode, variance & standard deviation

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

MEAN – average value of the data set

symbol(s): 𝜇 − 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛

𝑥̅ – sample mean,

𝑥̅ = 𝑥1 + 𝑥2 + ...+𝑥𝑛 = ∑𝑛𝑖 = 1 𝑥𝑖

𝑛

𝑛

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

𝑥̅ = ∑5𝑖 = 1 𝑥𝑖 = 𝑥1 + 𝑥2+𝑥3+𝑥4+𝑥5 = 1 + 4 + 4 + 6 + 10 = 25 = 5

5

5

5

5

MEDIAN – “the middle number”, 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 don’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 – 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, …)
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 𝑥 and population mean 𝜇 , 𝑥 – 𝜇 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): σ2 – population variance s2 – sample variance

σ2 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛
s2 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛−1

STANDARD DEVIATION – square root of the variance

Symbol(s): σ – population standard deviation

σ = √σ2 = √∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛

s – sample standard deviation

s = √s2 = √∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛−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 – Compute the sample mean 𝑥̅ Step 2 – Calculate the difference of 𝑥𝑖 − 𝑥̅ , for each value in the data set Step 3 – Calculate the squared difference (𝑥𝑖 − 𝑥̅)2, for each value in the data set Step 4 – Sum the squared differences ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2 Step 5 – Divide the sum of squared differences with n-1, 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑠2 = ∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛−1
Step 6 – ONLY FOR STANDARD DEVIATION Calculate the squared root of variance, 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = √𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒
s = √∑𝑛𝑖=1(𝑥𝑖 − 𝑥̅)2
𝑛−1

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