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Aditya 10th B (3)
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- 1Introduction to Statistics Ungrouped Data Ungrouped data is data in its original or raw form. The observations are not classified into groups. For example, the ages of everyone present in a classroom of kindergarten kids with the teacher is as follows: 3, 3, 4, 3, 5, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 27. This data shows that there is one adult present in this class and that is the teacher. Ungrouped data is easy to work when the data set is small.Grouped Data
In grouped data, observations are organized in groups.
For example, a class of students got different marks in a school exam. The data is tabulated as follows:
Mark interval 0-20 21-40 41-60 61-80 81-100
No. of Students 13 9 36 32 10
This shows how many students got the particular mark range. Grouped data is easier to work with when a large amount of data is present. - 2Frequency Frequency is the number of times a particular observation occurs in data. Class Interval Data can be grouped into class intervals such that all observations in that range belong to that class. Class width = upper class limit – lower class limitMean
Finding the mean for Grouped Data when class Intervals are not given
For grouped data without class intervals,
Mean = x¯=∑xifi∑fi where fi is the frequency of ith observation xi.
Finding the mean for Grouped Data when class Intervals are given
For grouped data with class intervals,
Mean = x¯=∑xifi∑fi
Where fi is the frequency of ith class whose class mark is xi.
Classmark =(Upper Class Limit+ Lower Class Limit)/2 - 3Direct method of finding mean Step 1: Classify the data into intervals and find the corresponding frequency of each class. Step 2: Find the class mark by taking the midpoint of the upper and lower class limits. Step 3: Tabulate the product of class mark and its corresponding frequency for each class. Calculate their sum (∑xifi). Step 4: Divide the above sum by the sum of frequencies (∑fi) to get the mean. Assumed mean method of finding mean Step 1: Classify the data into intervals and find the corresponding frequency of each class. Step 2: Find the class mark by taking the midpoint of the upper and lower class limits. Step 3: Take one of the xi’s (usually one in the middle) as assumed mean and denote it by ′a′. Step 4: Find the deviation of ′a′ from each of the x′is di=xi−a Step 5: Find the mean of the deviations d¯=∑xidi∑fi Step 6: Calculate the mean as x¯=a+∑xidi∑fiStep-Deviation method of finding mean
Step 1: Classify the data into intervals and find the corresponding frequency of each class.
Step 2: Find the class mark by taking the midpoint of the upper and lower class limits.
Step 3: Take one of the x′is (usually one in the middle) as assumed mean and denote it by ′a′.
Step 4: Find the deviation of a from each of the x′is
di=xi−a
Step 5: Divide all deviations −di by the class width (h) to get u′is.
ui=xi−ah
Step 6: Find the mean of u′is
u¯=∑fiui∑fi
Step 7: Calculate the mean as
x¯=a+h×∑fiui∑fi=a+hu¯ - 4Important relations between methods of finding mean All three methods of finding mean yield the same result. Step deviation method is easier to apply if all the deviations have a common factor. Assumed mean method and step deviation method are simplified versions of the direct method.Median
Finding the Median of Grouped Data when class Intervals are not given
Step 1: Tabulate the observations and the corresponding frequency in ascending or descending order.
Step 2: Add the cumulative frequency column to the table by finding the cumulative frequency up to each observation.
Step 3: If the number of observations is odd, the median is the observation whose cumulative frequency is just greater than or equal to (n+1)/2
If the number of observations is even, the median is the average of observations whose cumulative frequency is just greater than or equal to n/2 and (n/2)+1. - 5Cumulative Frequency Cumulative frequency is obtained by adding all the frequencies up to a certain point. Finding median for Grouped Data when class Intervals are given Step 1: find the cumulative frequency for all class intervals. Step 2: the median class is the class whose cumulative frequency is greater than or nearest to n2, where n is the number of observations. Step 3: Median=l+n2−cff×h Where, l=lower limit of median class, n= number of observations, cf= cumulative frequency of class preceding the median class, f= frequency of median class, h= class size (assuming class size to be equal).Cumulative Frequency distribution of less than type
Cumulative frequency of the less than type indicates the number of observations which are less than or equal to a particular observation.
Cumulative Frequency distribution of more than type
Cumulative frequency of more than type indicates the number of observations which are greater than or equal to a particular observation. - 6Visualising formula for median graphically Statistics for class 10 -1 Median from Cumulative Frequency Curve Step 1: Identify the median class. Step 2: Mark cumulative frequencies on the y-axis and observations on the x-axis corresponding to the median class. Step 3: Draw a straight line graph joining the extremes of class and cumulative frequencies. Step 4: Identify the point on the graph corresponding to cf=n/2 Step 5: Drop a perpendicular from this point on to the x-axis.Ogive of less than type
The graph of a cumulative frequency distribution of the less than type is called an ‘ogive of the less than type’.
Ogive of more than type
The graph of a cumulative frequency distribution of the more than type is called an ‘ogive of the more than type’.
To know more about Ogive, visit here.
Relation between the less than and more than type curves
The point of intersection of the ogives of more than and less than types gives the median of the grouped frequency distribution. - 7Prepared by :- Aditya Kumar Sharma Subject :-Maths. Topic :- Statistics Roll no. 10203 Class . :- 10th B.Thank-you
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- 10th grade Math Flashcards on Aditya 10th B (3), created by Aditya Sharma 10 th B on 11/07/2021.
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