Maharashtra State Board · Class 10 · Mathematics · Algebra Chapter 6

Statistics — Formula Sheet

Board Formulas

16 formulas

1.Class Mark
$x_{i} = \frac{Lower class limit + Upper class limit}{2}$
The class mark represents the whole class in mean calculations. For 20–30 the class mark is 25.
2.Mean: Direct Method
$X ˉ = \frac{\sum f _{i} x _{i}}{\sum f _{i}}$
$X ˉ$ : Mean of the data · $x_{i}$ : Class mark of the i-th class · $f_{i}$ : Frequency of the i-th class
Multiply each class mark by its frequency, add, and divide by the total frequency. Fine for small numbers; heavy arithmetic for large ones.
3.Deviation from Assumed Mean
$d_{i} = x_{i} - A$
A is the assumed mean, usually a class mark near the middle. dᵢ is negative for classes below A.
4.Mean: Assumed Mean Method★
$d ˉ = \frac{\sum f _{i} d _{i}}{\sum f _{i}}, X ˉ = A + d ˉ$
$A$ : Assumed mean · $d_{i}$ : Deviation of class mark from A · $d ˉ$ : Mean of the deviations
Smaller numbers than the direct method, and the answer is the same whatever A you choose.
5.Step Deviation
$u_{i} = \frac{x _{i} - A}{g}$
The textbook takes g as the G.C.D. of all the dᵢ; for equal classes with A at a class mark, g is the class width. Then the uᵢ are small integers like −2, −1, 0, 1, 2.
6.Mean: Step Deviation Method★
$u ˉ = \frac{\sum f _{i} u _{i}}{\sum f _{i}}, X ˉ = A + g u ˉ$
$u_{i}$ : Step deviation of the i-th class · $g$ : G.C.D. of all dᵢ (usually the class width) · $u ˉ$ : Mean of the step deviations
Do not forget to multiply ū by g at the end.
7.Making Classes Continuous
$Correction = \frac{( lower limit of next class ) - ( upper limit of a class )}{2}$
Subtract the correction from every lower limit and add it to every upper limit. For 10–19, 20–29, … the correction is 0.5, giving 9.5–19.5, 19.5–29.5, …
8.Cumulative Frequency (less than type)
$c f_{k} = f_{1} + f_{2} + \dots + f_{k}$
Running total of frequencies. The last cumulative frequency equals N.
9.Median Class
$Median class = first class with c f \geq \frac{N}{2}$
Find N/2, then go down the cf column until it first reaches or crosses N/2.
10.Median of Grouped Data★
$Median = L + [\frac{\frac{N}{2} - c f}{f}] \times h$
$L$ : Lower class limit of the median class · $N$ : Total frequency (Σfᵢ) · $c f$ : Cumulative frequency of the class preceding the median class · $f$ : Frequency of the median class · $h$ : Class width
cf is the cumulative frequency of the class BEFORE the median class, not of the median class itself.
11.Modal Class
$Modal class = class with the highest frequency$
Find it before applying the mode formula; f₀ and f₂ are the frequencies of the classes just before and just after it.
12.Mode of Grouped Data★
$Mode = L + [\frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}}] \times h$
$L$ : Lower class limit of the modal class · $f_{1}$ : Frequency of the modal class · $f_{0}$ : Frequency of the class preceding the modal class · $f_{2}$ : Frequency of the class succeeding the modal class · $h$ : Class width
If the modal class is the first class, take f₀ = 0; if it is the last class, take f₂ = 0.
13.Frequency Polygon
$Plot the points (x_{i}, f_{i})$
Join the points (class mark, frequency) by straight lines. Close the polygon on the X-axis at the class marks of the classes just before the first class and just after the last class (frequency 0). It can also be drawn by joining the mid-points of the tops of the histogram rectangles.
14.Pie Diagram: Central Angle★
$θ = \frac{Value of the component}{Total value} \times 36 0^{\circ}$
Each component gets a sector whose angle is proportional to its share of the total.
15.Pie Diagram: Value from the Angle
$Value of the component = \frac{θ}{36 0 ^{\circ}} \times Total value$
Used when the pie diagram is given and you have to find the data.
16.Check for a Pie Diagram
$\sum θ = 36 0^{\circ}$
If your angles do not add up to 360°, recheck the calculation before drawing.