Descriptive Statistics Calculator

Have a look at the simple and easy guidelines on how to find the descriptive statistics for sample and population data sets.

• Let us take either sample or population data set.
• Know the various formulas and substitute those values in the formulas.
• Do the required math calculations to find the answer.

Descriptive statistics are brief descriptive coefficients that summarize a data set, which can be either representation of the entire or a sample of a population. It includes mean, median, mode, minimum, maximum, range, sum, size, count, standard deviation, variance, midrange, quartiles, outliers, sum of squares, mean absolute deviation, root mean square, standard error of the mean, skewness, kurtosis, coefficient of variation, relative standard deviation.

Let us say the data set is x1, x2, x3, x4, . . . xn.

1. Minimum

Order the data set from lowest to highest i.e x1 ≤ x2 ≤ x3 ≤ x4 ≤ . . . xn

Minimum value = x_min = min(xi)_i=1ⁿ

2. Maximum

Order the data set from lowest to highest i.e x1 ≤ x2 ≤ x3 ≤ x4 ≤ . . . xn

Maximum value = x_max = max(xi)_i=1ⁿ

3. Range

Range of data set is defined as the difference between maximum and minimum.

Range = x_max - x_min

4. Sum

Sum is the total of all data items in the data set.

Sum = ∑_{i = 1}ⁿxi

5. Size, Count

Size or count is defined as the number of values in the data set.

Size = n = count(xi)_{i = 1}ⁿ

6. Mean

For population data set

μ = (∑_{i = 1}ⁿxi) / n = Sum / size

For sample data set

x_mean = (∑_{i = 1}ⁿxi) / n

7. Median

Order the given data from lowest to highest value. Median is the value that separate first half of the ordered sample data from the other half. If n is odd then median is the mid value. If n is even then median is the average of center 2 vales.

If n is odd the median value at position p is

p = (n + 1) / 2

x_median = x_p

If n is even, median becomes the average of middle two values at p and p+1 positions

p = n/2

x_median = ( x_p + x_p+1) / 2

8. Mode

Mode is the values that occur most frequently in the data set. A data set can have more than one mode or no mode.

9. Standard Deviation

For a population

σ = √[(∑_{i = 1}ⁿ (xi - μ)²) / n]

For a sample

s = √[(∑_{i = 1}ⁿ (xi - x_mean)²) / (n - 1)]

10. Variance

For a population

σ² = [(∑_{i = 1}ⁿ (xi - μ)²) / n]

For a sample

s² = [(∑_{i = 1}ⁿ (xi - x_mean)²) / (n - 1)]

11. Relative Standard Deviation

For a population

RSD = [(100 * σ) / μ]%

For a sample

RSD = [(100 * s) / x_mean]%

12. Midrange

It is the average of minimum and maximum values.

MR = (x_min + x_max) / 2

13. Quartiles

Quartiles separates given data set into four sections. Median is the second quartile Q2, which divides ordered data set into two half's. First quartile Q1 is the median of the lower half that not includes Q2. Third quartile Q3 is the median of the higher half not including Q2.

14. Interquartile Range

Range from Q1 to Q3 is the interquartile range (IQR)

IQR = Q3 - Q1

15. Outliers

Upper Fence = Q3 + 1.5 x IQR

Lower Fence = Q1 - 1.5 x IQR

16. Sum of Squares

For a Population

SS = ∑_{i = 1}ⁿ (xi - μ)²

For a Sample

SS = ∑_{i = 1}ⁿ (xi -x_mean)²

17. Mean Absolute Deviation

For a Population

MAD = (∑_{i = 1}ⁿ |xi - μ|) / n

For a Sample

MAD = (∑_{i = 1}ⁿ |xi -x_mean|) / n

18. Root Mean Square

RMS = √[(∑_{i = 1}ⁿ (xi)²) / n ]

19. Standard Error of the Mean

For a Population

SE_μ = (σ / (√n))

For a Sample

SE_xmean = (s / (√n))

20. Skewness

For a Population

γ1 = (∑_{i = 1}ⁿ (xi - μ)³) / (n * σ³)

For a Sample

γ1 = [(n) / ((n - 1) (n - 2)] [∑_{i = 1}ⁿ ((xi - x_mean) / s)³]

21. Kurtosis

For a Population

β2 = (∑_{i = 1}ⁿ (xi - μ)⁴) / (n * σ⁴)

For a Sample

β2 = [(n (n +1)) / ((n - 1) (n - 2) (n - 3)] [∑_{i = 1}ⁿ ((xi - x_mean)⁴ / s)]

22. Kurtosis Excess

For a Population

α4 = [((∑_{i = 1}ⁿ (xi - μ)⁴) / (n * σ⁴)) - 3]

For a Sample

α4 = {[(n (n + 1)) / ((n - 1)(n - 2)(n - 3)] [∑_{i = 1}ⁿ((xi - x_mean) / s)⁴] - [(3(n-1)²) / ((n - 2) (n - 3))]

23. Coefficient of Variation

For a Population

CV = σ / μ

For a Sample

CV = s / x_mean

Example

Question: Calculate the descriptive statistics of sample data set {1, 8, 56, 15, 9, 25}

Solution:

Given sample data set is {1, 8, 56, 15, 9, 25}

Order of the given data set is {1, 8, 9, 15, 25, 59}

Minimum:

Minimum = min(xi)_i=1ⁿ

Min = x1 = 1

Maximum

Maximum = max(xi)_i=1ⁿ

Max = x6 = 59

Range

Range = x_n - x1

= 59 - 1 = 58

Sum

Sum = ∑_{i = 1}ⁿxi

= (1 + 8 + 56 + 15 + 9 + 25)

= 117

Size

Size = n = count(xi)_{i = 1}ⁿ

= 6

Mean

x_mean = (∑_{i = 1}ⁿxi) / n

= (1 + 8 + 56 + 15 + 9 + 25) / 6

= 117/6 = 19.5

Median

p = n/2 = 6/2 = 3

x_median = ( x_p + x_p+1) / 2

= (9 + 15) / 2 = 24/2 = 12

Mode

No mode

Standard Deviation

s = √[(∑_{i = 1}ⁿ (xi - x_mean)²) / (n - 1)]

= √[[(1 - 19.5)² + (8 - 19.5)² + (9 - 19.5)² + (15 - 19.5)² + (25 - 19.5)² + (59 - 19.5)²] / (6-1)]

= √[[(-18.5)² + (-11.5)² + (-10.5)² + (-4.5)² + (5.5)² + (39.5)² / 5]

= √[[342.25 + 132.25 + 110.25 + 20.25 + 30.25 + 1560.25] / 5]

= √(2195.5 / 5)

= √439.1

Standard Deviation = 20.95

Variance

s² = [(∑_{i = 1}ⁿ (xi - x_mean)²) / (n - 1)]

= [[(1 - 19.5)² + (8 - 19.5)² + (9 - 19.5)² + (15 - 19.5)² + (25 - 19.5)² + (59 - 19.5)²] / (6-1)]

= [[(-18.5)² + (-11.5)² + (-10.5)² + (-4.5)² + (5.5)² + (39.5)² / 5]

= [[342.25 + 132.25 + 110.25 + 20.25 + 30.25 + 1560.25] / 5]

= (2195.5 / 5)

Variance = 439.1

Mid Range

MR = (x_min + x_max) / 2

= (1 + 59) / 2 = 60/2

Mid Range = 30

Quartiles

According to the definition

Q1 is 8

Q2 is 12

Q3 is 25

Interquartile Range

IQR = Q3 - Q1

= 25 - 8

= 17

Outliers

Upper Fence = Q3 + 1.5 x IQR

= 25 + 1.5 x 17

= 50.5

Lower Fence = Q1 - 1.5 x IQR

= 8 - 1.5 x 17

= -17.5

Sum of Squares

SS = SS = ∑_{i = 1}ⁿ (xi -x_mean)²

= (∑_{i = 1}ⁿ (xi - x_mean)²)

= (1 - 19.5)² + (8 - 19.5)² + (9 - 19.5)² + (15 - 19.5)² + (25 - 19.5)² + (59 - 19.5)²

= (-18.5)² + (-11.5)² + (-10.5)² + (-4.5)² + (5.5)² + (39.5)²

= 342.25 + 132.25 + 110.25 + 20.25 + 30.25 + 1560.25

= 2195.5

Mean Absolute Deviation

MAD = (∑_{i = 1}ⁿ |xi -x_mean|) / n

= |(1 - 19.5) + (8 - 19.5) + (9 - 19.5) + (15 - 19.5) + (25 - 19.5) + (59 - 19.5)| / 6

= |(-18.5) + (-11.5) + (-10.5) + (-4.5) + (5.5) + (39.5)| / 6

= |-18.5 - 11.5 - 10.5 - 4.5 + 5.5 + 39.5| / 6

= |-45 + 45| / 6

= 0

Root Mean Square

RMS = √[(∑_{i = 1}ⁿ (xi)²) / n ]

= √[(1² + 8² + 56² + 15² + 9² + 25²) / 6]

= √[(1 + 64 + 3136 + 225 + 81 + 625) / 6

= √[4132 / 6]

= √688.66

RMS = 26.24

Standard Error of the Mean

SE_xmean = (s / (√n))

= 20.954 / √6

= 20.954 / 2.44

Standard Error of the Mean = 8.58

Skewness

γ1 = [(n) / ((n - 1) (n - 2)] [∑_{i = 1}ⁿ ((xi - x_mean) / s)³]

= [(6) / ((6 - 1) (6 - 2)] x [((1 - 19.5) / 20.954)³ + ((8 - 19.5) / 20.954)³ + ((9 - 19.5) / 20.954)³ + ((15 - 19.5) / 20.954)³ + ((25 - 19.5)/ 20.954)³ + ((59 - 19.5) / 20.954)³]

= [ (6) / ((5) * (4))] x [(-18.5 / 20.954)³ + (-11.5 / 20.954)³ + (-10.5 / 20.954)³ + (-4.5 / 20.954)³ + (5.5 / 20.954)³ + (39.5 / 20.954)³]

= [6 / 20] x [(-0.882)³ + (-0.548)³ + (-0.501)³ + (-0.214)³ + (0.262)³] + (1.885)³]

= [6 / 20] x [-0.686 - 0.164 - 0.125 - 0.009 + 0.017 + 6.697]

= [0.3] x 5.73

= 1.719

Kurtosis

β2 = [(n (n +1)) / ((n - 1) (n - 2) (n - 3)] [∑_{i = 1}ⁿ ((xi - x_mean)⁴ / s)]

= [(6 (6 +1)) / ((6 - 1) (6 - 2) (6 - 3)] x [((1 - 19.5) / 20.954)⁴ + ((8 - 19.5) / 20.954)⁴ + ((9 - 19.5) / 20.954)⁴ + ((15 - 19.5) / 20.954)⁴ + ((25 - 19.5)/ 20.954)⁴ + ((59 - 19.5) / 20.954)⁴

= [(6 x 7) / (5 x 4 x 3)] x [(-18.5 / 20.954)⁴ + (-11.5 / 20.954)⁴ + (-10.5 / 20.954)⁴ + (-4.5 / 20.954)⁴ + (5.5 / 20.954)⁴ + (39.5 / 20.954)⁴]

= [35 / 60] x [(-0.882)⁴+ (-0.548)⁴ + (-0.501)⁴ + (-0.214)⁴ + (0.262)⁴] + (1.885)⁴]

= (0.583) x [0.605 + 0.0901 + 0.063 + 0.00209 + 0.0047 + 12.62]

= 0.583 x 16.080

= 9.3758

Kurtosis Excess

α4 = {[(n (n + 1)) / ((n - 1)(n - 2)(n - 3)] [∑_{i = 1}ⁿ((xi - x_mean) / s)⁴] - [(3(n-1)²) / ((n - 2) (n - 3))]

= {[(6 (6 + 1)) / ((6 - 1)(6 - 2)(6 - 3)] x [((1 - 19.5) / 20.954)⁴ + ((8 - 19.5) / 20.954)⁴ + ((9 - 19.5) / 20.954)⁴ + ((15 - 19.5) / 20.954)⁴ + ((25 - 19.5)/ 20.954)⁴ + ((59 - 19.5) / 20.954)⁴] - ((3(6 - 1)²) / ((6 - 2) (6 - 3))]

= {[(6 x 7) / (5 x 4 x 3)] x [(-18.5 / 20.954)⁴ + (-11.5 / 20.954)⁴ + (-10.5 / 20.954)⁴ + (-4.5 / 20.954)⁴ + (5.5 / 20.954)⁴ + (39.5 / 20.954)⁴] - [(3(5)²) / ((4) (3))]}

= {[42 / 60] x [(-0.882)⁴+ (-0.548)⁴ + (-0.501)⁴ + (-0.214)⁴ + (0.262)⁴] + (1.885)⁴] - [(3 x 25) / 12]}

= {(0.7) x [0.605 + 0.0901 + 0.063 + 0.00209 + 0.0047 + 12.62] - (75 / 12)}

= {0.7 x 13.38 - 6.25}

= {9.375 - 6.25}

= 3.125

Coefficient of Variation

CV = s / x_mean

= 20.954 / 19.5

CV = 1.07

Relative Standard Deviation

RSD = [(100 * s) / x_mean]%

= [(100 * 20.954) / 19.5]%

RSD = 107.45%

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1. What are the four types of descriptive statistics?

The four different types of descriptive statistics are measures of frequency, measures of central tendency, measures of dispersion or variation and measures of position.

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2. What is meant by descriptive statistics?

Descriptive statistics describes the characteristics of a data set. It contains two basic categories of measures. They are measure of central tendency describes the center of a data set and measure of variability describe the dispersion of data within the set.

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3. What are the main methods of descriptive statistics?

The three main types of descriptive statistics are the frequency distribution, variability of a dataset and central tendency.

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4. What are the different types of statistics?

Two different types of statistical methods which are used in analyzing the data are descriptive statistics and inferential statistics.

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