| --- |
| source: ncert |
| topic: statistics |
| class_level: class_10 |
| chapter: ch14 |
| difficulty: intermediate |
| --- |
| |
| Class 10 | Ch14: Statistics |
| MEAN FOR GROUPED DATA: |
| Direct method: x_bar = sum(f*x)/sum(f). x=class mark=(upper+lower)/2. |
| Assumed mean method: x_bar = a + sum(f*d)/sum(f). d=x-a. |
| Step deviation method: x_bar = a + (sum(f*u)/sum(f))×h. u=(x-a)/h. Best when h is constant. |
| MEDIAN FOR GROUPED DATA: |
| Formula: M = L + ((n/2 - cf)/f) × h. |
| L=lower boundary of median class. |
| n=total frequency. cf=cumulative frequency of class BEFORE median class. |
| f=frequency of median class. h=class width. |
| Median class: class where cumulative frequency first exceeds n/2. |
| MODE FOR GROUPED DATA: |
| Modal class=class with highest frequency. |
| Mode = L + ((f1-f0)/(2f1-f0-f2)) × h. |
| L=lower boundary of modal class. f1=frequency of modal class. |
| f0=frequency of class BEFORE modal class. f2=frequency of class AFTER modal class. |
| EMPIRICAL RELATIONSHIP: Mode = 3×Median - 2×Mean (approximately true). |
| OGIVE (CUMULATIVE FREQUENCY CURVE): |
| Less than ogive: plot (upper class boundary, cumulative frequency). |
| More than ogive: plot (lower class boundary, more than cumulative frequency). |
| Median from ogive: find n/2 on y-axis, draw horizontal to curve, then vertical to x-axis. |
| SOLVED EXAMPLES: |
| Example 1: Mean by step deviation. Classes 10-20,20-30,...50-60. Frequencies 4,6,8,5,3. a=30, h=10. |
| Class marks: 15,25,35,45,55. u=(-2,-1,0,1,2). f×u=-8,-6,0,5,6. |
| sum(f)=26. sum(f×u)=-3. Mean=30+(-3/26)×10=30-1.15=28.85. |
| Example 2: Find median. n=26, n/2=13. cf values: 4,10,18,23,26. 13 falls in class 30-40 (cf=10, f=8). |
| Median=30+((13-10)/8)×10=30+3.75=33.75. |
| COMMON MISTAKES: |
| For median formula: cf is cumulative frequency of class BEFORE (not including) median class. |
| Class mark is MIDPOINT: for class 10-20, mark=15 (not 10 or 20). |
| Modal class: highest frequency (not highest class mark). |
