Statistics Formulas

Solving various kinds of questions related to statistics can be done easily by using these basic Statistics Formulas. So, get the important formulae list from this section and get familiar with the calculations at a faster pace. The list of formulas are as mentioned here

1. Mean (Arithmetic Mean)

Mean or arithmetic mean is also called as average. It is defined as the sum of of data points divided by the count. Perform the sum of all data values in the sample and divide it by the number of data only. Population mean is expressed as μ where as sample mean is expressed as x with a dash at the top symbol. The formulas of population mean and sample mean are given below

For Sample Data Set

x_mean = (∑_{i = 1}ⁿx_i) / n = [x₁ + x₂ + x₃ + . . . + x_n] / n

For Population Data Set

μ = (∑_{i = 1}ⁿx_i) / n = Sum / size = [x₁ + x₂ + x₃ + . . . + x_n] / n

2. Geometric Mean

It is defined as the product of all data values raised to the power of the reciprocal of count of data values. Multiply data values of the sample and find the power of reciprocal of count of sample. Simply, nth root of product of data values is geometric mean.

GM = (Π_{i = 1}ⁿ)^1/n = (n)√(x₁.x₂.x₃. . . . x_n)

3. Harmonic Mean

Harmonic mean is n times the reciprocal of the sum of the reciprocals of the data values. Or, Divide n by the sum of reciprocals of all data values.

HM = (n) / [∑_{i = 1}ⁿ(1/x_i)] = (n) / [1/x₁ + 1/x₂ + 1/x₃ + . . . + 1/x_n

4. Median

Median is the middle of the sorted list of data values. Order all values of the data set from lowest to highest value i.e ascending order, x₁ ≤ x₂ ≤ x₃ ≤ . . . ≤ x_n. In other way, median is a numerical number which separate's the ordered data set into two half's. If n is the odd number then median is middle value. If n is an even number then median is the average of center 2 values.

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

p = n/2

x_{median> = (xp + xp+1) / 2}

If n is odd, median is the middle number

p = (n + 1) / 2

x_{median> = xp}

5. Mode

Mode is a number or numbers that appears most often in the data set.

6. Weighted Mean | Weighted Average

Weighted mean is the sum of weights times the mean of data set divided by the sum of weights. Its formula is as follows

x_{weighted_mean} = ∑_{i = 1}ⁿ(w_i * x_mean) / ∑_{i = 1}ⁿw_i

7. Minimum

Write the data set in the ascending order. Then, minimum value is the first value or x₁

x₁ ≤ x₂ ≤ x₃ ≤ . . . ≤ x_n

Minimum = x₁ = min(x_i)_i=1ⁿ

8. Maximum

Ordering a data set from lowest to the highest value, maximum is the largest value which is x_n.

x₁ ≤ x₂ ≤ x₃ ≤ . . . ≤ x_n

Maximum = x_n = max(x_i)_i=1ⁿ

9. Range

It is obtained by finding the difference between maximum and minimum values of the data set. To find range, you must know x_n, x₁.

Range = R = Maximum - Minimum = x_n - x₁

10. Midrange

It is the average of minimum and max_imum values in the data set.

MR = (Minimum + Maximum) / 2 = (x₁ + x_n) / 2

11. Frequency

The number of occurrences for each data value in the data set is called frequency.

12. Count

Count is the total number of values in the data set. This is also known as the size of the data set.

Count = n = count(x_i)_{i = 1}ⁿ

13. Sum

The total of all data values in the data set is sum.

sum = x₁ + x₂ + x₃ + . . . + x_n = ∑_{i = 1}ⁿx_i

14. Percentile

Arrange all data values in ascending order i.e x₁ ≤ x₂ ≤ x₃ ≤ . . . ≤ x_n

Find the rank for the percentile p using r = (p/100) * (n - 1) + 1

If r is an integer then, data value at the r location is percentile p

p = x_r

If r is not a integer, p is interpolated using ri, the integer part of r and rf, the fractional part of r

p = x_ri + r_f * (x_ri+1 - x_ri)

15. Quartiles

Quartiles are used to divide data sets into four equal parts of data values. The median is the second quartile Q₂. It divides an ordered data set into upper & lower halves. The first quartile Q₁ is the median of the lower half not including Q₂. The third quartile Q₃ is the median of the upper half not including Q₂.

16. Interquartile Range

The range from Q1 to Q3 is the interquartile range IQR.

IQR = Q₃ - Q₁

17. Outliers

Outliers are values that lie above the upper fence or below the lower fence of a data set. The outliers formulas are used to find potential outliers within a sample data set.

Upper Fence = Q₃ + 1.5 x IQR

Lower Fence = Q₁ - 1.5 x IQR

18. Sum of Squares

It is the sum of squared difference between data value and mean.

For Population Data Set

SS = ∑_{i = 1}ⁿ (x_i - μ)²

For Sample Data Set

SS = ∑_{i = 1}ⁿ (x_i -x_mean)²

19. Standard Deviation

Standard deviation is a measure of the dispersion of data values from the mean. Its formula is

For a Population Data Set

σ = √[(∑_{i = 1}ⁿ (x_i - μ)²) / n]

For a Sample Dara Set

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

20. Variance

Variance is calculated as the sum of squared deviation of each data value from the mean, divided by the data sample size. It is a measure of dispersion of the data points from the mean. It is also called the square of standard deviation.

For a Population Data Set

σ² = [(∑_{i = 1}ⁿ (x_i - μ)²) / n]

For a Sample Data Set

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

21. Relative Standard Deviation

RSD is 100 times the standard deviation divided by the mean.

For a Population Data Set

RSD = [(100 * σ) / μ]%

For a Sample Data Set

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

22. Z-score

Z-score is a measure of how many standard deviations a data value is above or below the mean of a data set. It is calculated using single data point

z = (x - μ) / σ

When we have average of sample data set

z = (x_mean - μ) / σ/√n

23. Mean Deviation (Mean Absolute Deviation MAD)

It is the sum of absolute difference between data value and mean divided by count of the data set.

For a Population Data Set

MAD = (∑_{i = 1}ⁿ |x_i - μ|) / n

For a Sample Data Set

MAD = (∑_{i = 1}ⁿ |x_i -x_mean|) / n

24. Root Mean Square (RMS)

It is the square root of sum of squares of data values divided by the size of the sample data.

RMS = √[(∑_{i = 1}ⁿ (x_i)²) / n ]

25. Standard Error of the Mean

Standard Error of the Mean is obtained by dividing the standard deviation with the square root of size of the data sample.

For a Population Data Set

SE_μ = (σ / (√n))

For a Sample Data Set

SE_xmean = (s / (√n))

26. Skewness

Skewness Formula for both population and sample data sets are given here

For a Population Data Set

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

For a Sample Data Set

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

27. Coefficient of Variation

It is the standard deviation divided by mean.

For a Population Data Set

CV = σ / μ

For a Sample Data Set

CV = s / x_mean

28. Kurtosis

For a Population Data Set

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

For a Sample Data Set

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

29. Kurtosis Excess

For a Population Data Set

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

For a Sample Data Set

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

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