__Statistics__

**Find the mean number of plants per house. Statistics deals with collection, presentation, analysis and interpretation of numerical data.**

**Arranging data in a order to study their salient features is called presentation of data.**

**Range of the data is the difference between the maximum and the minimum values of the observations**

**Table that shows the frequency of different values in the given data is called a frequency distribution table**

**A frequency distribution table that shows the frequency of each individual value in the given data is called an ungrouped frequency distribution table.**

**A table that shows the frequency of groups of values in the given data is called a grouped frequency distribution table**

**The groupings used to group the values in given data are called classes or class-intervals. The number of values that each class contains is called the class size or class width. The lower value in a class is called the lower class limit. The higher value in a class is called the upper class limit.**

**Class mark of a class is the mid value of the two limits of that class.**

**Mean of Grouped Data**

**The mean (or average) of observations, as we know, is the sum of the values of all the observations divided by the total number of observations**

**Mean x̄ **

**= (f**_{1}**x**_{1}** + f**_{1}**x**_{1}** + L + f**_{n}**x**_{n}**) **

** f**_{1}** + f**_{1}** + L + f**_{n}** **

**Mean x̄ **

**= ∑ f**_{i}**x**_{i}**/f**_{i}

**where i varies from 1 to n**

**We can form ungrouped data into grouped data by forming class-intervals of some width.**

**It is assumed that the frequency of each classinterval is centred around its mid-point.**

**Class mark **

**= (Upper class limit + Lower class limit)/2**

**Direct method **

**Mean x̄ **

**= ∑ f**_{i}**x**_{i}**/f**_{i}

**Assumed Mean method **

**Mean x̄ **

**= a + [∑ f**_{i}**d**_{i}**]/f**_{i}

**We can only locate a class with the maximum frequency, called the modal class. The mode is a value inside the modal class, and is given by the formula:**

**Mode**

**= l + [(f**_{1}** - f**_{0}**)/((2f**_{1}**) - f**_{0}** - f**_{2}**)]*h**

**where l = lower limit of the modal class,**

**h = size of the class interval (assuming all class sizes to be equal),**

**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.**

**for finding the median of ungrouped data, we first arrange the data values of the observations in ascending order. Then, if n is odd, the median is the (n+1)/2 th observation. And, if n is even, then the median will be the average of the n/2 th and (n/2)+1 th observations**

**Less than cumulative frequency distribution:**

**It is obtained by adding successively the frequencies of all the previous classes including the class against which it is written. The cumulate is started from the lowest to the highest size.**

**More than cumulative frequency distribution:**

**It is obtained by finding the cumulate total of frequencies starting from the highest to the lowest class.**

**To find median class, we find the cumulative frequencies of all the classes and n/2. We now locate the class whose cumulative frequency is greater than (and nearest to) n/2.**

**After finding the median class, we use the following formula for calculating the median.**

**Median **

**= l + [((n/2) - cf)/f]*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).**

**There is a empirical relationship between the three measures of central tendency :**

**3 Median = Mode + 2 Mean**

**Representing a cumulative frequency distribution graphically as a cumulative frequency curve, or an ogive of the less than type and of the more than type.**

**The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for this data.**

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