Tales by Dots and Lines – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash Part II, tailored for Class 8 in Mathematics.
This one-pager compiles key formulas and equations from the Tales by Dots and Lines chapter of Ganita Prakash Part II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Mean (Arithmetic Mean): μ = (x₁ + x₂ + ... + xₙ) / n
μ represents the mean, x₁, x₂, ..., xₙ are the data values, and n is the number of values. The mean is a measure of central tendency that summarizes a set of data points by a single value.
Median: M = (n is odd) -> x((n+1)/2) or M = (x(n/2) + x((n/2)+1))/2 (n is even)
M is the median, n is the total number of values. It represents the middle value of a data set when arranged in ascending order. If n is odd, the median is the middle value; if even, it's the average of the two middle values.
Finding Missing Data Value: x = n * μ - Σxi
x is the missing value, n is the total number of data points, μ is the mean, and Σxi is the sum of the known data points. This formula allows for the calculation of a value when the mean and the other values are known.
Effect of Adding Value: μ' = (Σxi + k) / (n + 1)
μ' is the new mean after adding k. This shows how including a new value k can affect the mean, illustrating balance in data adjustments.
Effect of Removing Value: μ' = (Σxi - k) / (n - 1)
μ' is the new mean after removing k. It demonstrates how the mean changes when a value is taken out, highlighting the relevance of data points to mean calculation.
Increased Collection: μ' = μ + c
μ' is the new mean after each data point is increased by a constant c. This shows that adding a constant raises the mean by that constant's value.
Doubled Collection: μ' = 2μ
μ' represents the mean when all data points are multiplied by 2. The mean also doubles, illustrating proportional relationships in multiplication.
Frequency Mean: Mean = (Σ(f * x)) / Σf
f is the frequency of each value x. This formula calculates the mean considering the number of occurrences of each value, essential for grouped data analysis.
Grouping Data for Median: Position = (N + 1) / 2
N is the total number of data points. This formula helps to find the position of the median in grouped frequency tables.
Median in Frequencies: M = (Lower Class Limit + (h(f(N/2 - C))) / f)
Where h is the interval size, C is cumulative frequency just before the median class. This provides a way to calculate median from grouped data in frequency tables.
Equations
Coconut Harvest: 25.6 = z / 15
z is the total number of coconuts. This equation helps find total harvest based on average and number of trees.
Weight Calculation: (42 + 40 + 39 + 33 + 48 + 38 + 42 + 35 + 32 + w) / 10 = 39.2
This equation finds the unknown weight w of the players, using the average.
Average Family Size: (3×3 + 4×11 + 5×9 + 6×7 + 7×3 + 8×1 + 9×1 + 10×1) / 36 = 5.22
This shows how to calculate the average family size considering repeated values through frequency.
New Average after Correction: 381 / 15 = 25.4
This equation calculates the correct average harvest per tree after correcting one tree's harvest.
Determine Positions for Median: Cumulative Frequency = 14 for value 4, 23 for value 5
This illustrates how to use cumulative frequencies to identify median positions without exhaustive listing.
Mean Calculation with Inclusion: μ' = (Σxi + 11) / 11
Shows how inclusion of a value (11) alters the mean when total values are 11.
Two Values Inclusion Impact: 0 < x < μ: μ' = constant
This shows that including values above and below the mean can stabilize mean results.
Grouping Method for Median: M = (n + 1)/2
Helps identify median's position in a data set by counting values.
Data Shift Calculation: μ' = μ + 10
This highlights mean displacement when all data points are incremented.
Balance of Distances: |X - μ| = |Y - μ|
This relation indicates how values relate around the mean, identifying the central tendency.
