Provided Descriptive Statistics Calculator is useful to calculate the Relative Standard Deviation, Standard Deviation, Variance, Skewness, Mean, Median, Mode, others for both sample and population data set. Just enter data set in the input section, select whether it is sample or population data set and hit on the calculate button to get the each and every statistics as output with an brief explanation in the output section.

**Descriptive Statistics Calculator: **Are you stuck at some point while calculating the descriptive statistics of the data set? Then Don't worry as you have arrived the right place. You can use our best Descriptive Statistics Calculator tool to get the exact result easily without doing any math calculations. Go through the entire article to know the complete details about statistics.

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=1}^{n}

** 2. Maximum**

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

Maximum value = x_{max} = max(xi)_{i=1}^{n}

** 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}^{n}xi

** 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}^{n}xi) / n = Sum / size

For sample data set

x_{mean} = (∑_{i = 1}^{n}xi) / 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 * σ^{3})

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 * σ^{4})

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 * σ^{4})) - 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=1}^{n}

Min = x1 = 1

** Maximum**

Maximum = max(xi)_{i=1}^{n}

Max = x6 = 59

** Range**

Range = x_{n} - x1

= 59 - 1 = 58

** Sum**

Sum = ∑_{i = 1}^{n}xi

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

= 117

** Size**

Size = n = count(xi)_{i = 1}^{n}

= 6

** Mean**

x_{mean} = (∑_{i = 1}^{n}xi) / 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%

Learn more about the other mathematical calculators that provides instant results from Onlinecalculator.guru

**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.

**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.

**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.

**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.