ICSE Class 10 Maths Chapter 29 Median Quartiles and Mode

Chapter 29

Median, Quartiles and Mode

Points To Remember

1 Median: The value of the middle-most observation obtained after arranging the data in an ascending order, is called median of the data.

2 Median of an Ungrouped Data:

Method: Arrange the data in ascending order. Let the total number of observations be N.

(i) If N is odd, then Median = \(\left(\frac{N+1}{2}\right)\) th observation.

(ii) If N is even, then Median is the average of \(\left(\frac{N}{2}\right)\) th and \(\left(\frac{N}{2}+1\right)\) th observations.

3 Quartiles: The observations which divide the whole set of observations into four equal parts, are known as quartiles.

4 Lower Quartile (or First Quartile): If the variates are arranged in ascending order, then the observation lying midway between the lower extreme and the median is called the lower quartile, or First quartile, to be denoted by Q₁.

5 Upper Quartile (or Third Quartile): If the variates are arranged in ascending order, then the observation lying midway between the median and the upper extreme is called the upper quartile or Third quartile, to be denoted by Q₃. For an ungrouped data containing n observations, we have:

Lower Quartile, Q₁ = \(\begin{cases} \left(\frac{N}{4}\right) \text{ th observation, if N is even;} \\ \left(\frac{N+1}{4}\right) \text{ th observation, if N is odd} \end{cases}\)

Upper Quartile, Q₃ = \(\begin{cases} \left(\frac{3N}{4}\right) \text{ th observation, if N is even;} \\ \frac{3(N+1)}{4} \text{ th observation, if N is odd.} \end{cases}\)

Remark: The Middle quartile is the Median, to be denoted by Q₂.

Teacher's Note

Understanding quartiles helps us analyze data distribution - just like dividing test scores into quarters helps identify which students are in the top 25% of the class.

6 Inter Quartile Range: (Q₃ - Q₁) is called the interquartile range.

7 Semi-Interquartile Range: \(\frac{1}{2}\) (Q₃ - Q₁) is called the semi-interquartile range.

8 Median and Quartiles from Ogive: If the given frequency distribution is not continuous, convert it into the continuous form. Prepare the Cumulative Frequency Table. Draw ogive for the cumulative frequency distribution given. Let, total number of observations = Sum of all frequencies = N.

9 For Estimating Median: Mark a point A on y-axis, corresponding to (N/2). Note that in graphical location, this value is taken to be (N/2), whether N is odd or even. From P, draw a vertical line PM to meet x- axis at M. Then, the abscissa of M gives the Median.

10 For Estimating Lower Quartile Q₁ and Upper Quartile Q₃:

Remarks: (i) To locate the value of Q₁ on ogive, we mark the point along y-axis, corresponding to (N/4) and proceed similarly.

(ii) To locate the value of Q₃ on ogive, we mark the point along y-axis, corresponding to (3N/4) and proceed similarly.

11 Mode: The value around which there is the greatest concentration is called the mode.

12 Mode For Individual Data: In the case of individual data, the mode is the variate which occurs most frequently.

13 Empirical Formula: We may calculate the mode by using the formula: Mode = 3 (Median) -2 (Mean)

14 Estimation of mode from Histogram:

(i) If the given frequency distribution is discontinuous, then convert it into the continuous form.

(ii) Draw a histogram to represent the above data.

(iii) From the upper corners of the highest rectangle, draw line segments to meet opposite corners of adjacent rectangles, diagonally. Let these line segments intersect at a point P.

(iv) Draw PM perpendicular on x-axis to meet it at M. Then the abscissa of M gives the mode of the data.

Teacher's Note

The mode is the most frequently occurring value - like the most popular shoe size in a store, which helps retailers stock their inventory effectively.

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