Skip to content
← Maths

Class 10 · Chapter 13 · Mathematics

Statistics Cheat Sheet

Formulae

Mean (direct method)

xˉ=∑fixi∑fi\bar{x} = \frac{\sum f_{i} x_{i}}{\sum f_{i}}

fᵢ = frequency of the class, xᵢ = class mark, Σfᵢ = total frequency.

Mean (assumed mean method)

xˉ=a+∑fidi∑fi,di=xi−a\bar{x} = a + \frac{\sum f_{i} d_{i}}{\sum f_{i}}, \qquad d_{i} = x_{i} - a

a = assumed mean, dᵢ = deviation of each class mark from a.

Mean (step-deviation method)

xˉ=a+h(∑fiui∑fi),ui=xi−ah\bar{x} = a + h\left(\frac{\sum f_{i} u_{i}}{\sum f_{i}}\right), \qquad u_{i} = \frac{x_{i}-a}{h}

h = class size, uᵢ = step deviation. Useful when class marks are large.

Median of grouped data

Median=l+n2−cff×h\text{Median} = l + \frac{\dfrac{n}{2} - cf}{f} \times h

l = lower limit of the median class, n = total frequency, cf = cumulative frequency before the median class, f = frequency of the median class, h = class size.

Mode of grouped data

Mode=l+f1−f02f1−f0−f2×h\text{Mode} = l + \frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} \times h

l = lower limit of the modal class, f₁ = frequency of the modal class, f₀ = frequency of the class before it, f₂ = frequency of the class after it, h = class size.

Empirical relationship

3 Median=Mode+2 Mean3\,\text{Median} = \text{Mode} + 2\,\text{Mean}

An approximate relation between the three measures of central tendency.

Definitions

Grouped data
Data arranged in classes or intervals with the frequency of each class.
Class mark
The midpoint of a class interval, equal to the average of its lower and upper limits.
Class size
The difference between the upper and lower limits of a class interval; denoted h.
Frequency
The number of observations falling in a particular class.
Cumulative frequency
The running total of frequencies up to a given class.
Mean
The average of the observations; the sum of all values divided by their number.
Median
The middle value of the data when arranged in order; it divides the data into two equal halves.
Mode
The value that occurs most frequently in the data.

Key Points

  • Ungrouped data can be converted to grouped data by forming classes and counting frequencies.
  • The mean of grouped data can be found by the direct, assumed-mean or step-deviation method; all give the same value.
  • The class mark xᵢ is taken as the representative value of a class interval.
  • The median class is the class whose cumulative frequency first becomes greater than or equal to n/2.
  • The modal class is the class with the highest frequency.
  • The three measures of central tendency are mean, median and mode; each describes the data in a different way.
  • For a symmetrical distribution the mean, median and mode are nearly equal, and the empirical relation 3 Median = Mode + 2 Mean holds approximately.