# Statistical Methods-Measures of Location (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 7 of 7

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## Question number: 1

» Statistical Methods » Measures of Location

Appeared in Year: 2010

### Describe in Detail

A cyclist pedals from his house to his college at a speed of l0 km per hour and back from the college to his house at 15 km per hour. Find the average speed.

### Explanation

Given that a cyclist pedals from his house to his college at a speed of l0 km per hour and back from the college to his house at 15 km per hour. Assume that the distance between the house to college is d. Here the average speed is finding by harmonic mean because it is suitable when the values are pertaining to the rate of change per unit time su…

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## Question number: 2

» Statistical Methods » Measures of Location

Appeared in Year: 2010

### Describe in Detail

Let F _{n} (x) be the distribution function define by

Is the lim F _{n} (x) a distribution function? If not, why?

### Explanation

The function F _{n} (x) is a distribution function if

By the definition , but here this is not equal to one. This says that F _{n} (x) is not a distribution function.

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## Question number: 3

» Statistical Methods » Measures of Location

Appeared in Year: 2011

### Describe in Detail

Let Z be a random variable with p. d. f. f (z). Let z _{α} be its upper α ^{th} quantile. Show that if X is

a random variable with p. d. f. then σz _{α} +µ is the upper α ^{th} quantile of X.

### Explanation

Let Z be a random variable with p. d. f. f (z). Given that the quantile of z is defined as

X is a random variable with p. d. f. , then quantile is

Assume

Then the limit is

Lower limit is - and upper limit is

If z= then the integral is same for given z and the quantile is

So, the quantile of X is .

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## Question number: 4

» Statistical Methods » Measures of Location

Appeared in Year: 2010

### Describe in Detail

Show that for any discrete distribution, standard deviation is not less than mean deviation from mean.

### Explanation

Here, we show that standard deviation is greater than mean deviation from mean.

S. D. ≥ mean deviation from mean

(S. D. ) ^{2} ≥ (mean deviation from mean) ^{2}

Let x _{1}, x _{2}, …, x _{n} are n observations in a set have frequency f _{1}, f _{2, } …, f _{n} and the mean of observation is , then

Assume

Which is essentially true as f _{1}, f _{2}, … are not ne…

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## Question number: 5

» Statistical Methods » Measures of Location

Appeared in Year: 2015

### Describe in Detail

Let X have the probability density function

Find the mean, geometric mean and harmonic mean.

### Explanation

Then mean is

The geometric mean is

The geometric mean is

The harmonic mean is

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## Question number: 6

» Statistical Methods » Measures of Location

Appeared in Year: 2013

### Describe in Detail

Find the weighted arithmetic mean of the first ‘n’ natural numbers, the weights being the corresponding numbers.

### Explanation

If x _{1}, x _{2}, …, x _{n} are the values having weights w _{1}, w _{2}, …, w _{n} respectively, the weight mean is

Here X is first ‘n’ natural numbers that is 1, 2, 3, …, n

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## Question number: 7

» Statistical Methods » Measures of Location

Appeared in Year: 2013

### Describe in Detail

For the following frequency distribution on 229 values:

Class | Frequency |

0 - 10 | 12 |

10 - 20 | 30 |

20 - 30 | x |

30 - 40 | 65 |

40 - 50 | y |

50 - 60 | 25 |

60 - 70 | 18 |

the median is found to be 46. Find the values of x and y.

### Explanation

The cumulative frequency is

Class | Frequency | Cumulative frequency |

0 - 10 | 12 | 12 |

10 - 20 | 30 | 42 |

20 - 30 | x | 42 + x |

30 - 40 | 65 | 107 + x |

40 - 50 | y | 107 + x+y |

50 - 60 | 25 | 132 + x+y |

60 - 70 | 18 | 150 + x+y |

Median is 46 which lies in the class 40 - 50. So, median cla…

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