Here, CF (🞁) = Cumulative Frequency (Ascending) and,
CF (▼) = Cumulative Frequency (descending).

Arithmetic mean (AM) = \( \frac{\sum f_i \times x_i}{N} \)
Geometry mean (GM) = \( \text{antilog} \left( \frac{\sum f_i \log(x_i)}{N} \right) \)
Harmonic mean (HM) = \( \frac{N}{\sum f_i / x_i} \)
For two non-zero positive observations, \( AM \geq GM \geq HM \)
and, \( (AM \times HM) = GM^2 \)

\( L_m = \text {Lower boundary of median class} \)
\( F_m = \text {Cumulative frequency of the class before median class} \)
\( f_m = \text {Frequency of median class} \)
\( h = \text {Class width or, interval} \)
Median = \( L_m + \left( \frac{\frac{N}{2} - F_m}{f_m} \right) \times h \)

\( i_{th} \) Quartiles = \( {i_{th} \times N} \over 4 \)th observed class ; divide the distribution into 4 equal parts.
\( Q_{i} = L_i + {{{iN \over 4}-F_i} \over {f_i}} \times h ; i = 1, 2, 3 \)
\( i_{th} \) Deciles = \( {i_{th} \times N} \over 10 \)th observed class ; likewise, 10 equal parts.
\( D_{j} = L_j + {{{jN \over 10}-F_j} \over {f_j}} \times h ; j = 1, 2,...9 \)
\( i_{th} \) Percentiles = \( {i_{th} \times N} \over 100 \)th observed class ; likewise, 100 equal parts.
\( P_{k} = L_k + {{{kN \over 100}-F_k} \over {f_k}} \times h ; k = 1, 2,...99 \)
Here, \( L_i \) = Lower boundary of the quantile class
\( F_i \) = Cumulative frequency of the class before quantile class
\( f_i \) = Frequency of quantile class

Modal class = Class with the highest frequency
Mode = \( L_m + \left( \frac{f_0 - f_{1}}{2f_0 - f_{1} - f_{2}} \right) \times h \)
where, \( L_m \) = Lower boundary of the modal class