Statistics and Probability

**Category : **10th Class

**Statistics and Probability**

**Statistics**

Statistics is the branch of Mathematics which deals with the collection and interpretation of data. The data may be represented in different graphical forms such as bar graphs, histogram, ogive curve, and pie chart. This representation of data reveals certain salient features of the data. These values of the data are called measure of central tendency. The various measures of central tendencies are mean, median and mode. A measure of central tendency gives us the rough idea of where data points are centered. But in order to make more accurate interpretation of central values of the data, we should also have an idea of how the data are scattered around the measure of central tendency.

**Mean Deviation about Mean of an Ungrouped Data**

Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\]be the n observations, then the mean of the data is given by:

\[\overline{x}=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+---+{{x}_{n}}}{n}\]\[\Rightarrow \overline{x}=\frac{1}{n}\sum\limits_{k\,=\,1}^{n}{{{X}_{k}}}\]

Then the deviation of the data from the mean is given by: \[|{{x}_{1}}-\overline{x}|,|{{x}_{2}}-\overline{x}|,|{{x}_{3}}-\overline{x}|,---|{{x}_{n}}-\overline{x}|\]

Now the mean deviation of the data is given by \[\frac{1}{n}\sum\limits_{k=1}^{n}{|{{X}_{k}}-\overline{x}|}\]

**Mean Deviation about Mean of a Grouped Data**

Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\]be the n - observations and \[{{f}_{1}},\text{ }{{f}_{2}},\text{ }{{f}_{3}},---,\text{ }{{f}_{n}}\]be the corresponding frequencies of the data. Then the mean of the data is given by: \[\overline{x}=\frac{{{x}_{1}}{{f}_{1}}+{{x}_{2}}{{f}_{2}}+---+{{x}_{n}}{{f}_{n}}}{{{f}_{1}}+{{f}_{2}}+---+{{f}_{n}}}\] or, \[\overline{x}=\frac{\sum\limits_{k\,=\,1}^{n}{{{X}_{k}}}{{f}_{k}}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\]

Then the mean deviation about mean is given by \[\frac{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}|{{x}_{k}}-\overline{x}|}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\]

**Mean Deviation About Median of an Ungrouped Data**

The median of an ungrouped data is obtained by arranging the data in the ascending order. If the data contains odd number of terms, then the median is \[{{\left( \frac{n+1}{2} \right)}^{th}}\]I term of the data and if the data contains even number of terms, then the median is the average of \[{{\left( \frac{n}{2} \right)}^{th}}\]and\[{{\left( \frac{n}{2}+1 \right)}^{th}}\]terms i.e., \[\frac{{{\left( \frac{n}{2} \right)}^{th}}term+{{\left( \frac{n}{2}+1 \right)}^{th}}term}{2}\].

If M is the median of the data, then mean deviation about M is given by \[\frac{1}{n}\sum\limits_{k\,=\,1}^{n}{|{{x}_{k}}-M}|\]

Mean Deviation About Median of a Grouped Data

Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\] be the n -observations and \[{{f}_{1}},\text{ }{{f}_{2}},\text{ }{{f}_{3}},---,\text{ }{{f}_{n}}\] be the corresponding frequencies of the data. Then the mean deviation about the median of the data is given by: \[\frac{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}|{{x}_{k}}-M|}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\]

For the grouped data the median can be obtained by \[l+\left( \frac{\frac{N}{2}-C}{f} \right)\times h\]

Where, I = lower limit of the median class

N = sum of all frequencies

c = cumulative frequency of preceding median class

h = class width

f = frequency of the median class

**Standard Deviation and Variance**

Standard deviation is the square root of the arithmetic mean of the squares of deviations of the terms from their arithmetic mean and it is denoted by o. The square of standard deviation is called the variance.

Thus for simple distribution, \[\sigma =\sqrt{\frac{\sum\limits_{i\,=\,1}^{n}{{{({{x}_{1}}-\overline{x})}^{2}}}}{n}}\]

**Note**:

(i) The standard deviation of any arithmetic progression is \[\sigma =\,\,|d|\sqrt{\frac{{{n}^{2}}-1}{12}}\]where d = common difference and n = number of terms of the A.P.

(ii) Coefficient of variation (C.V.) =\[\frac{\sigma }{x}\]\[\times \]100

**Probability**

We have studied about the probability as a measure of uncertainty of various phenomenon in our daily life. Normally we obtain the probability of an event as the ratio of the number of favourable outcomes to that of the total possible outcomes. This is called classical theory of probability. Up to class IXth we have studied classical theory of probability. But the drawback of this theory is that it cannot be applied to the activities which have infinite numbers of outcomes. In this chapter we will study about this approach which is called axiomatic approach of probability.

**Random Experiment**

An experiment is said to be a random experiment if it has more than one possible outcomes and it is not possible to predict the outcome in advance. For example, throwing a die or tossing a coin is a random experiment.

**Outcomes**

The possible results of a random experiment are called outcomes.

For example, throwing a dice and getting 1, 2, 3, 4, 5, 6 are the possible outcomes.

**Sample Space**

The set of all possible outcomes of a random experiment is called sample space. Each element of the sample space is called sample point. For example, when we throw a die we get {1, 2, 3, 4, 5, 6}, then it is called sample space.

**Sample Space for a Pair of Dice**

When two dice are thrown then the sample space is:

{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.

**A Deck of Cards**

We know that there are 52 cards in a deck of cards, it has four types, which are spade, club, heart and diamond. Ail are equally divided. It means there are 13 spades, 13 clubs, 13 hearts and 13 diamonds. The colour of spade and club cards are black and the colour of heart and diamond cards are red. Thus in 52 cards, 26 are red cards and 26 are black cards. King, queen and jack are called face cards, the total number efface cards is 12, in which 6 are red face cards and 6 are black face cards.

**Events**

The subset E of a sample space is called an event. There are different types of events which are given below:

**Impossible Events**

The events which will never occur is called impossible events. The empty set is also called impossible events, for example throwing a dice and getting 7 is an impossible event.

**Sure Events**

The events which will definitely occur are called sure events. Throwing a dice and getting 1, 2, 3, 4, 5, or 6 is a sure event

Mutually Exclusive Events Two or more events associated with a random experiment are said to be mutually exclusive events, if the occurrence of any one of them prevents the occurrence of all others.

**Probability of Events**

For any two events A and B, the probability of union of the two sets is given by: \[P\,(A\bigcup B)=P(A)+P(B)-P(A\bigcap B)\]

If A and B are disjoints then \[A\bigcap B=\phi ,\] then \[P(A\bigcup B)=P(A)+P(B)\]

If A and B are independent events then, \[P(A\bigcap B)=P(A)\times P(B)\]

Thus, \[P(A\bigcup B)=P(A)+P(B)-P(A)\times P(B)\]

- P (not A) = 1 ? P (a)
- P (A but not B) \[=P(A\bigcap B')=P(A)-P(A\bigcap B)\]
- P (neither A nor B) \[=P(A'\,\bigcap B')=P(A\bigcup B)'=1-P(A\bigcup B)\]
- Example:

In a box, there are 18 red, 11 blue and 15 green balls. One ball is picked up randomly. What is the probability that it is either red or green?

(a) \[\frac{1}{4}\] (b) \[\frac{1}{2}\]

(c) \[\frac{3}{4}\] (d) \[\frac{5}{4}\]

(e) None of these

Ans. (c)

Explanation: Here, total number of balls = 18 + 11 + 15 = 44 balls.

Let E = Event that the ball drawn is either red or green

\[\therefore n(E)=18+15=33\] \[\therefore P(E)=\frac{33}{44}=\frac{3}{4}\]

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