Measures of Dispersion : Range , Quartile Deviation, Standard Deviation and Variance.

Measures of Dispersion: 

I. Introduction.

II. Requirements of good measures.

III. Uses of Measures of Dispersion.

IV. Methods Of Studying Dispersion:

    i. Absolute Measures of Dispersions: 

         i. Range (R) 

        ii. Quartile Deviation (Q.D.) 

        iii. Mean Deviation (M.D.)

        iv. Standard Deviation (S. D.)

        v. Variance

   ii. Relative Measures of Dispersions: 

         i. Coefficient of Range 

        ii. Coefficient of Quartile Deviation (Q.D.) 

        iii. Coefficient of Mean Deviation (M.D.)

        iv. Coefficient of Standard Deviation (S. D.)

        v. Coefficient of Variation (C.V.) 

                                                                                                              

I. Introduction.

We have the various measures of central tendency, like Mean, Median & Mode,  it is a single figure that represent the whole data. Now we are interested to study this figure(i.e. measures of central tendency) is proper representative of  actual values of the data. If most of the actual values in the data are close to the average then it is properly represent the data, and if the actual values of data are away from the average then it not properly represent the data. In that case we are interested to study  how far the actual values away from the average is known as Dispersion. i.e. Dispersion Means the Spread of actual values from the average or mean. 

foe example we consider the following data, 

Observations					Total	Mean
Set A :	101	98	99	102	400	100
Set B :	1	0	1	398	400	100

The size of data and mean of the bot sets are same, is 100, but question is the mean is good representative of data or not ? both set have same mean, but the values in the set A are very close to the mean, then the mean is good representative, but in Set B the values are far from the mean. hence mean is not good representative of the data in set B, from this we see the difference between mean and actual values of data are less then mean is properly represent the data, (for set A), therefore we measure the variation in the data. or the measure of  Dispersion. 

    i. Definition:- Dispersion is the Spread of value from the mean, Or deviation of different values of the data from it mean is known as Dispersion.

Following are the Objectives of  Measures of Dispersion. 

    i. To Measure the Reliability of an average.

    ii. To Compare the variability of different distribution. 

    iii, To control the variability. 

                                                                                                              

II. Requirements of good measures.

The main objective of the Dispersion is to measure the Reliability of an Average. following are the Properties of good measures of dispersion.

        i. It should be simple to understand and rigidly defined.

        ii. It should be easy to calculate. 

        iii. It should be based on all observation in data.

        iv. It should have sampling stability.

        v. It should not ne unduly affected by extreme values. 

                                                                                                              

III. Uses of Measures of Dispersion.

Measure of Dispersion is also known as measure of variability or spread. following are the some uses of Measures of Dispersion.

    1.Understanding the distribution of data: Measure of dispersion help to understand the how data points are spread out around the central tendency (measure of central  tendency means mean , median , mode), a small dispersion indicates the data points are close to central value, while the dispersion is larger indicates the large variability in data set. 

    2. Comparing Data set: Using the measures of dispersion finding which data set has greater variability, or comparing this data set based on variability in data points.

                                                                                                              

IV. Methods Of Studying Dispersion:

There are two types of measures of dispersion, i) Absolute Measure of dispersion and ii) Relative Measures of dispersion. 

    i) Absolute Measures of Dispersions: The measure of dispersion is expressed in the term of original unit of the data are called Absolute Measures of Dispersion. and following are the Absolute measures of Dispersions. 

        i. Range (R) 

        ii. Quartile Deviation (Q.D.) 

        iii. Mean Deviation (M.D.)

        iv. Standard Deviation (S. D.)

        v. Variance

    ii.)  Relative Measures of Dispersions: The measure of dispersion is expressed in Ratio or Percentage of are called Relative Measures of Dispersion. and following are the Relative measures of Dispersions. 

        i. Coefficient of Range 

        ii. Coefficient of Quartile Deviation (Q.D.) 

        iii. Coefficient of Mean Deviation (M.D.)

        iv. Coefficient of Standard Deviation (S. D.)

        v. Coefficient of Variation (C.V.) 

                                                                                                              

we study one by one the measures of Dispersions 

I. Range

Definition:- The range is the one of the simplest method of measuring Dispersion. It is defined as the Difference between the largest and smallest values of the data. Or the range is defined as the difference between maximum and minimum values in data sets. 

 it is formulated as

Range = L - S

Where L:- the largest value in data

S:- the smallest value in data. 

Sometimes the Range is denoted as R.

It is an absolute measure of Dispersion. 

Now the Relative measure of Dispersion corresponding to Range is called the coefficient of range. It is gives by 

Coefficient of range = (L-S)/(L+S)

📑 MERITS OF RANGE

I. It is simple to understand.

II. It is easy to calculate. 

📑DEMERITS OF RANGE

I. It has no sampling stability.

II. It is effected by extreme values.

Range is the simplest measure that it gives quick idea about the spread in data. a large range indicates a wider variability, and a smaller range indicates narrower Spread. (i.e. the data points in dataset  are closer together and it has less variability) it indicates less dispersion. the greatest drawback of range is that is too much affected by extreme values.

Examples on Range: 

1. Calculate the range of sales revenue for a company over a month.

Day’s:	1	2	3	4	5	6
Sales Revenue:	500	700	600	910	250	800

 also find coefficient of range.

solution:

for finding the range we need to determining the difference between the Largest and Smallest values in data.

the largest revenue is 910 and smallest revenue is 250

Range = Largest  revenue value - Smallest revenue value

Range = 910 - 250

Range = 660

And coefficient of Range = ( Largest  revenue value - Smallest revenue value) / ( Largest  revenue value + Smallest revenue value)

Coefficient of Range = (910-250) / (910+250)

Coefficient of Range = 660 / 1160

Coefficient of Range =  0.5689

therefore The Range = 660 and coefficient of Range = 0.5689

 {it is example of discrete data set}

Range for continuous Data or distribution. 

if the data is continuous distribution then the Range is calculates as Difference between the upper limit of  the highest class limits and the lower limit of the Lowest class limits. (i.e. difference between upper limit of  last class and lower limit of the first class)

2. Find the Range and Coefficient of range  for the following data.

I.Q.	60-70	70-80	80-90	90-100	100-110	110-120
Freq.	7	12	28	42	30	10

Solution:  for finding the range we need to determining the Difference between the upper limit of  the highest class limits and the  lower limit of the Lowest class limits. 

the upper limit of  the highest class limits = 120

and  the  lower limit of the Lowest class limits = 60

Range = 120-60  = 60

Range = 60

and coefficient of range = (120-60) / (120+60)   = 60 / 180 = 0.3333

The Range = 60 and Coefficient of Range = 0.3333

                                                                                                              

II. Quartile Deviation: 

 we already seen that the greatest drawback of range is that is too much affected by extreme values. this drawback can be overcome by ignoring the extreme value, this can be done by ignoring 25% observation then finding the range of the data. that mean difference between third quartile and first quartile. this range is called Quartile Deviation (Q.D.) Or semi-inter quartile range. it is denotes as Q.D.

it is given by 

Quartile Deviation  = ( Q₃ – Q₁) / 2

Where Q₃ is the third quartile of the data. 

and  Q₁is the first quartile of the data.

and the quartile are calculates as Q₁  = size of {(N+1) / 4 } th item.

here N  is the number of observations in data.

   Q₃=size of {[3(N+1)] / 4 } th item.

the Q.D. is absolute measure of dispersion and the coefficient of Q.D. is the relative measure of dispersion.

therefore the coefficient of Q.D. =  ( Q₃ – Q₁)  /  ( Q₃ + Q₁) 

📑 MERITS OF Quartile Deviation

1. it is simple to understand 

2. easy to calculate.

3. it is not effected by extreme values.

4. it is specially useful in case of open-end classes.

📑DEMERITS OF Quartile Deviation

1.it is not useful for further mathematical calculations.

2. it ignoring the first 25% and last 25% data.

3. it is not based on all observations.

4. it has not sampling stability.

The quartile deviation is ignoring first 25% and last 25% data. mean it is based on the middle 50% data.

a large quartile deviation indicates the greater dispersion means grater variation in dataset, while the smaller quartile deviation indicates the smaller variation in data set.

 we see the example of Quartile Deviation And Coefficient of Quartile Deviation in next post. the Quartile Deviation is Based on the First and Third Quartiles. 

Quartile Deviation:

Quartile Deviation id calculated for i. Individual data, ii. Discrete data, iii. Continuous data

i. Individual Data: 

Example 1. Calculating the quartile deviation and coefficient of quartile deviation for the following data.

17, 20, 35,  51,  28,  14,  11. 

Solution: 

for calculating the quartile deviation and coefficient of quartile deviation firstly arranging the data in ascending in order 11,  14,  17,  20,  28,  35,  51.

then for calculating quartile deviation is calculated as 

Quartile Deviation =  (Q₃ – Q₁)/2

where  Q₁ = Size of {(n+1)/4}th item.

and  Q₃ = Size of {3(n+1)/4}th item.

 Q₁ = Size of {(n+1)/4}th item. =  size of  {(7+1)/4} th item. = size of (8/4) th observation

 Q₁ = Size of  2 nd observation =   14 

 Q₁ = 14

 Q₃ = Size of {3(n+1)/4}th item. =  Size of {3(7+1)/4}th item. = Size of {3(8)/4}th item. 

Q₃ = size of (3x 2) th observation = size of 6 th observation 

Q₃ = 35.

Quartile Deviation =  (Q₃ – Q₁)/2  = (35-14)/2  = 21/2

.Quartile Deviation = 10.5

Coefficient of Quartile Deviation =  (Q₃ – Q₁)/ (Q₃ + Q₁)

Coefficient of Quartile Deviation =  21/ 49   = 0.4285

Coefficient of Quartile Deviation = 0.4285

therefore the Coefficient of Quartile Deviation is0.4285 and Quartile Deviation is 10.5

ii. For Discrete Distribution 

Example 2. Calculating the quartile deviation and coefficient of quartile deviation for the following data.

Marks	20	30	40	50	60
No. of Students	2	13	7	8	1

Solution: firstly we arranging the data in ascending order and adding cumulative frequency column.

Marks	No. of Students F	Cumulative Frequency C.F.
20	2	2
30	13	15
40	7	22
50	8	30
60	1	31

Q₁ = Size of {(n+1)/4}th item.

and  Q₃ = Size of {3(n+1)/4}th item.

Q₁ = Size of {(n+1)/4}th item. =  size of  {(31+1)/4} th item. = size of (32/4) th observation

 Q₁ = Size of  8 th observation =   30 

  because 8 th observation in corresponding to 30 marks i.e. 1to 15 observation (means 15 students have marks less than or equal to 30)corresponding to 30 and 8 is lies between 1 to 15

Q₃ = Size of {3(n+1)/4}th item. =  Size of {3(31+1)/4}th item. = Size of {3(32)/4}th item. 

Q₃ = Size of 24 th observation = 50 

. because 30students have marks less than or equal to 50 then 24 th number student have mark 50         ( because 22 students have marks less than or equal to 40 therefore 23 have mark 50)

  Quartile Deviation =  (Q₃ – Q₁)/2   = (50-30) /2  =  10

 Quartile Deviation is 10

Coefficient of Quartile Deviation =  (Q₃ – Q₁)/ (Q₃ + Q₁)

Coefficient of Quartile Deviation = 20/ 80   =  0.25

Coefficient of Quartile Deviation  = 0.25

therefore the Quartile Deviation is 10 and Coefficient of Quartile Deviation is  0.25

iii. For Continuous Distribution: 

Example 3. Calculating the quartile deviation and coefficient of quartile deviation for the following data.  

Marks	Frequency
60-70	2
70-80	7
80-90	12
90-100	28
100-110	42
110-120	36
120-130	18
130-140	10
140-150	3
150-160	2

Solution: firstly we arranging the data in ascending order and adding cumulative frequency column.

Marks	Frequency	C.F.
60-70	2	2
70-80	7	9
80-90	12	21
90-100	28	49
100-110	42	91
110-120	36	127
120-130	18	145
130-140	10	155
140-150	3	158
150-160	2	160
Total	160

Q₁ = Size of {(n)/4}th item.

and  Q₃ = Size of {3(n)/4}th item.

Q₁ = Size of {(n)/4}th item. =  size of  {(160)/4} th item. =  40 th observation

for 40 th observation is corresponding to class 90-100, and lower limit of first quartile is 90 

Q₁  = lower limit of first + {([n/4]-C.F.)/f} x i

Where C.F. = cumulative frequency of  previous class i.e. C.F. = 21

n/4 = 40

f = frequency of first quartile class i.e. f = 28

i = class width  =  upper limit - lower limit = 70-60 = 10

Q₁  = 90 + {(40-21)/28} x 10

Q₁  = 90+6.78 = 96.78

 Q₃ = Size of {3(n)/4}th item.= 120 th observation

the third quartile class is 110-120, lower limit of third quartile class  is 110

C.F. = cumulative frequency of  previous class i.e. C.F. = 91

i = class width  =  upper limit - lower limit = 70-60 = 10

f = frequency of first quartile class i.e. f =36

3[n/4] = 3[160/4] = 120

Q₃  = lower limit of first + {(3[n/4]-C.F.)/f} x i

Q₃  =  110 + ({120-91}/36) X 10 

Q₃  =  110 +8.06

Q₃  =  118.06 

Quartile Deviation  =  (Q₃ – Q₁)/2  = (118.06 - 96.78)/2

Quartile Deviation = 10.64

Coefficient of Quartile Deviation =  (Q₃ – Q₁)/ (Q₃ + Q₁)

Coefficient of Quartile Deviation = 21.28/ 214.84   =  0.0990

Coefficient of Quartile Deviation  = 0.0990

therefore the Quartile Deviation is 10.64 and Coefficient of Quartile Deviation is  0.0990

see all three examples to understand how to solved the examples based on the Quartile Deviation for individual, discrete and continuous data. the formula of Quartile Deviation based on the type or nature of data set, used proper formula to calculate Quartile Deviation And it coefficient of Quartile Deviation. 

III.  Mean Deviation:

We see the two measures of dispersions Range and Quartile Deviation, they both measure of dispersion not based on all observations, hence it is not correctly measure the how far the values from the average.  therefore we use new measure of dispersion that based on the all observations that is Mean Deviation hence it is superior than Range and Quartile Deviation. 

Mean Deviation is Defined as the arithmetic mean of the absolute Deviation of all the values from it central values,(central value mean measure of central tendency i.e. mean , median and mode). but we know that the sum of deviation taken from the median is minimum. so generally the deviation is taken from median. it is called Mean Deviation taken from Median, but in some cases the deviation is taken from Mean or Mode then it is called the Mean Deviation taken from Mean OR Mode. it denoted as M.D.

it is defined  as Mean Deviation  = (1/n) x ∑(|x_i – Median| ) if deviation taken from median.

 Mean Deviation  = (1/n) x ∑(|x_i – Mean| ) if deviation taken from Mean.

 Mean Deviation  = (1/n) x ∑(|x_i – Mode| ) if deviation taken from Mode.

where n is total number of observation in data set.

x_i  be the observations i = 1, 2 , ...............n

∑ is summation symbol 

the relative measure corresponding to Mean Deviation is called coefficient of Mean Deviation. and it is given as

Coefficient of Mean Deviation = (  Mean deviation ) / ( Median) 

📑 MERITS OF M.D. 

I. It is simple to understand.

II. It is easy to calculate. 

III. It is based on all observations.

IV. It is Less affected by end value.

📑DEMERITS OF M.D. 

I. It is not used for further calculations.

For calculating the Mean Deviation Follow these Steps.

i. First we finding the Mean of Data set by summing the all observations and dividing by total number of observations.

ii. for calculating the Mean Deviation we find for each observation the difference between the observation and mean. (if deviation is taken from mean).

iii. Adding the all absolute differences.

iv. Dividing the sum of absolute differences by the total number of observation in the data set.

it is Given as Mean Deviation  = (1/n) x ∑(|x_i – Median| ) if deviation taken from median.

If the Outliers are present in data set and the distribution is not symmetric in that case the Mean Deviation is used as alternative to Standard Deviation.

Mean Deviation is used in certain cases, but it is less commonly used as compared to other measures of dispersions such as the Standard Deviation or variance. the choice of the measure to use is depending on the specific requirements of the analysis and the nature of the data set.

The Mean Deviation is simple than other measure of dispersions. it is directly measure the average of deviation taken from the mean and it can be easily understood and easily interpretable.

Examples.

Example. 1. Calculate Mean Deviation and it coefficient for the following data.

Price in Rs:

150

512

240

270

300

240

180

Answer: we know that for finding mean deviation firstly we calculate mean  any one measure of central tendency i.e. Mean, Median, Mode then we calculating the Mean Deviation if deviation taken from mean or median or mode.

here we calculate Mean Deviation From Mean

therefore firstly calculating the mean

Mean = sum of all observation / (divided by ) total number of observations.

Mean = (150 + 512 + 240 + 270 + 300 + 240 +180) / 7

Mean = 270.2857

now  Mean Deviation  = (1/n) x ∑(|x_i – Mean| )

for deviation we create table as 

Price in           Rs:	|xi –Mean|
150	120.2857
512	241.7143
240	30.2857
270	0.2857
300	29.7143
240	30.2857
180	90.2857
Total	542.8571

∑|xi – Mean| = 542.8571
  Mean Deviation  = (1/n) x ∑(|x_i – Mean| )
 Mean Deviation  = (1/7) x 542.8571
 Mean Deviation  = 77.5510

and coefficient of M.D. = ( M.D.) / Mean

coefficient of M.D.  = 77.5510 / 270.2857

Coefficient of M.D. = 0.2869

therefore the Mean Deviation = 77.5510 and coefficient of M.D. = 0.2869 

Example. 2. Calculate Mean Deviation and it coefficient for the following data.

X:	10	15	20	25	30
F:	5	4	10	8	7

Answer: First we calculate mean and then deviation for each observations. and addition column F x [|xi – Mean|]

X:	F:	F x X	|xi – Mean|	F x [|xi – Mean|]
10	5	50	11.17645	55.88225
15	4	60	6.17645	24.7058
20	10	200	1.17645	11.7645
25	8	200	3.82355	30.5884
30	7	210	8.82355	61.76485
Total	34	720	31.17645	184.7058

Where ∑FX = 720 , ∑F = 34,  ∑F x (|x_i – Mean| ) = 184.7058, n = 34

Mean =  (∑FX) / (∑F)

Mean  = 720 / 34 

Mean = 21.1765

Mean Deviation  = (1/n) x ∑F x (|x_i – Mean| )

Mean  Deviation = (1/34) x 184.7058

Mean Deviation =  5.4325

And Coefficient of Mena Deviation = (M.D.) / Mean = 5.4325/ 21.1765  

Coefficient of Mean Deviation = 0.2565

Therefore the M. D. is 5.4325 and coefficient of mean deviation is 0.2565.

Example 3. Calculate Mean Deviation and it coefficient for the following data.

Class:	10-20	20-30	30-40	40-50	50-60	60-70
Frequency:	4	6	5	10	7	1

Answer: First we calculate mean and then deviation for each observations. and addition column F x [|xi – Mean|]

Class:	Frequency    F	Mid-Point   M	FM	|xi – Mean|	F x [|xi – Mean|]
20-Oct	4	15	60	23.9393	95.7572
20-30	6	25	150	13.9393	83.6358
30-40	5	35	175	3.9393	19.6965
40-50	10	45	450	6.0607	60.607
50-60	7	55	385	16.0607	112.4249
60-70	1	65	65	26.0607	26.0607
Total	33		1285	90	398.1821

Where ∑FM = 1285 , ∑F = 33, F x [|xi – Mean|] = 398.1821, n = 33

Mean =  (∑FM) / (∑F)

Mean  = 1285 / 33 

Mean = 38.9393

Mean Deviation  = (1/n) x F x [|xi – Mean|]

Mean  Deviation = (1/33) x 398.1821

Mean Deviation =  12.0661

And Coefficient of Mena Deviation = (M.D.) / Mean = 12.0661/ 38.9393

Coefficient of Mean Deviation = 0.3098

Therefore the M. D. is 12.0661and coefficient of mean deviation is 0.3098

 

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Calculate C.V.

Marks	0-10	10-20	20-30	30-40	40-50	50-60
F	17	27	36	47	24	12

Solution:

Marks	F	mid-point m	fm	fm^2
0-10	17	5	85	425
10 - 20	27	15	405	6075
20-30	36	25	900	22500
30-40	47	35	1645	57575
40-50	24	45	1080	48600
50-60	12	55	660	36300
Total	163		4775	171475
mean	29.29447853
S.D. =	13.92214548

C.V. = (S.D. / Mean) x 100

C.V. = ( 13.92214548 / 29.29447853) x 100

C.V. = 0.4752 X 100

C.V. = 47.52%

Standard Deviation

We see that some measures of dispersion include Range, Quartile Deviation, and Mean Deviation. However, these measures are not based on all the observations in a dataset, making them less reliable. Among all the measures of dispersion, Standard Deviation stands out as the most reliable and important measure. Standard Deviation is based on all observations in a dataset, making it more reliable than any other measure of dispersion. It is defined as the square root of the arithmetic mean of the square of deviations from the mean, often referred to as root mean square deviation. It is denoted as σ (Sigma) and written as S.D.

Standard Deviation Formula:

S.D. = σ = √((∑(xi - x̄)²) / n) = Square-root of [∑(xi - x̄)² / n]

OR

S.D. = σ = √((∑(xi²) / n) - (x̄)²)

Where xi is the ith observation (i = 1, 2, ..., n), and x̄ is the arithmetic mean of the observations or data.

Merits of Standard Deviation (S.D.)

• Easy to define: It is rigidly defined and simple to define.
• Based on all observations: It considers all observations in the dataset, making it responsive to changes in data.
• Sampling stability: It is less affected by sampling fluctuations compared to other measures of dispersion.
• Use in mathematical calculations and analysis.
• Reliable compared to other measures of dispersion.

Demerit of Standard Deviation (S.D.)

• Not always simple to understand and calculate.

Variance

The square of Standard Deviation is called Variance and is denoted as σ².

Variance Formula:

Variance (σ²) = (∑(xi - x̄)² / n) OR (∑(xi²) / n - (x̄)²)

Coefficient of Variation

The Coefficient of Variation (C.V.) is a measure that expresses variability as a percentage. It is used for comparing the variability of different groups and is calculated as:

Coefficient of Variation (C.V.) = ((S.D.) / Mean) x 100

The Coefficient of Standard Deviation is similar but not expressed as a percentage.

Difference between Variance and Standard Deviation

Variance measures how much the observations differ from the mean, and it is calculated by summing the squared differences and dividing by the number of observations. Standard Deviation is the square root of Variance and measures how much the observations spread from the mean, using the same units as the data. The key difference is that Variance measures the average squared distance from the mean, while Standard Deviation measures variability in the same units as the data.

the standard deviation is zero if all observations are equal, it is always positive and positive although the values are negative.

Standard Deviation and Variance Calculation for Individual Data

Calculation Method 1:

For calculating standard deviation and variance, we use the following formula:

S.D. = σ = √((∑((x_i-x̄)²))/n)

Variance = σ² = (∑((x_i-x̄)²))/n

Example 1:

Calculate the standard deviation and variance of the following data:

        52, 57, 59, 63, 68, 45, 59, 47, 57, 50, 60, 66, 40, 48, 54.
    

Answer:

Values (x)	(x-x̄)	(xi-x̄)2
52	-3	9
57	2	4
59	4	16
63	8	64
68	13	169
45	-10	100
59	4	16
47	-8	64
57	2	4
50	-5	25
60	5	25
66	11	121
40	-15	225
48	-7	49
54	-1	1

Total ∑x = 825

Total ∑((x_i-x̄)²) = 892

Total number of observations (n) = 15

Mean (x̄) = (∑x)/n = 55

Standard Deviation (S.D.) = σ = √(∑((x_i-x̄)²)/n) = √(892/15) = 7.7115

Variance (σ²) = (∑((x_i-x̄)²)/n = 59.4672

Calculation Method 2:

For calculating standard deviation and variance using the second formula:

S.D. = σ = √((∑(x_i²)/n)-(x̄)²)

Variance = σ² = (∑(x_i²)/n)-(x̄)²

Example 1:

Calculate the standard deviation and variance of the following data:

        52, 57, 59, 63, 68, 45, 59, 47, 57, 50, 60, 66, 40, 48, 54.
    

Answer:

Values (x)	x2
52	2704
57	3249
59	3481
63	3969
68	4624
45	2025
59	3481
47	2209
57	3249
50	2500
60	3600
66	4356
40	1600
48	2304
54	2916

Total ∑x = 825

Total ∑(x²) = 46267

Total number of observations (n) = 15

Mean (x̄) = (∑x)/n = 55

Standard Deviation (S.D.) = σ = √(∑(x²)/n-(x̄)²) = √(46267/15-(55)²) = 7.7115

Variance (σ²) = (∑(x²)/n-(x̄)²) = 59.4672

Standard Deviation and Variance Calculator

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