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Tag: Median

  • Quartiles, Deciles and Percentiles

    Quartiles, Deciles and Percentiles

    Quartiles divides a set of data into four equal parts.

    A median divides a set of data into two parts each with equal number of items.

    The first quartile, mostly referred to as the lower quartile contains 25% of the total data items. Lower quartile can be described as the median of the bottom half.

    Second quartile is actually the median of the whole data(50%).

    The third quartile is usually referred to as upper quartile and contains 75% of total data items. It can be described as the median of the upper half the data set.

    Formula for the getting the first quartile Q1

    Where

    • L is the lower class boundary of the quartile class.
    • n is the total frequency
    • c is the cumulative frequency above the quartile class
    • i is the class interval
    • f is the frequency of the lower quartile class
    Formula for the getting the second quartile Q2

    Second quartile Q2 is actually the median of the data

    it is calculated from:

    where

    • L is the lower class boundary of the median class.
    • n is the total frequency
    • c is the cumulative frequency above the median class
    • i is the class interval
    • f is the frequency of the median class
    Formula for the getting the third quartile Q3

    where

    • L is the lower class boundary of the upper quartile class.
    • n is the total frequency
    • c is the cumulative frequency above the third quartile class
    • i is the class interval of the upper quartile class
    • f is the frequency of the upper quartile class

    Deciles

    Deciles divides a set of data into ten equal parts.

    First decile is when n is divided by 10. that is; Decile = n/10

    where n is the total frequency for the data

    Percentiles

    Percentiles divides a set of data into hundred equal parts.

    one percentile is given as (1/100)*n

    In quartiles, deciles and percentiles, data is arranged in ascending order

    Example 5

    The table below shows the distribution of heights to the nearest cm of students in a school.

    Table of heights of some students

    Find (a) the median

    (b)(i) lower quartile (ii) upper quartile (iii) 80th percentile.

    Solution

    (a) The new frequency table for the data is shown here

    There are 130 students . Therefore, the median height is the 65th student. that is; median is 130/2.

    The 65th student falls in the 150-159 class. This class is called the median class.

    Using the formula for the median:

    (b) (i)

    Lower quartile Q1 = L + (n/4 – C)i/f, that is:

    ii)

    Upper quartile Q3= L + (3n/4)-23)*5/9

    (C)

    The 80th percentile of the data is given by 80/100)*130=104th value.

    The 104th student falls in the 160-169 class

    80th percentile= L+(80/100n-C)i/f

    The complete solution is as below:

    Example

    Determine the lower quartile and upper quartile for the following set of data

    15, 20, 16, 15, 18, 17, 13, 9, 17, 18, 11

    solution

    arranging in ascending order

    9, 11, 13, 15, 15, 16, 17, 17, 18, 18, 20

    The median number is 16. On left of 16 there are 5 values and on the right 5 values.

    16 is at the center of the data list

    9, 11, 13, 15, 15 | 16 | 17, 17, 18, 18, 20

    The first half contains: 9, 11, 13, 15, 15

    The central value in that lower half is 13 and it is the first quartile of the data

    The upper half includes: 17, 17, 18, 18, 20

    The central value is 18 and is hence the upper quartile for the data list

    Related Topics

  • Measures of central Tendency

    Measures of central Tendency

    Definition

    Measures of central tendency in statistics are single values that can be derived from the data set such that it can be used as the representative of the whole data.

    The most common measures of central tendency includes:

    • Mean
    • mode
    • Median

    Mean

    Mean is the average value for the data set. It is obtained by finding the total sum of all the values in the data set then divining it by the total frequency.That is;

    Mean = (Sum of all Values)/(Total frequency)

    consider the following set of data represents marks scored by a group of students in a math test:

    68,65,59,30,42,45,46,59,80,23,54, 45,54,30

    The sum of values in the data= 68+65+59+30+42+45+46+59+80+23+54+45+54+30=700

    The total frequency of the data set = 14 because there are 14 results in the test

    Mean = 700/14 = 50

    Mean value could be interpreted to mean the value that could result if data set was modified so that each item will have the same quantity.

    Example Problem

    10 athletes measured their masses which were recorded and their mean determined as 60.45 Kg. The mass for nine of them were 62.10kg, 58.90kg,56.8kg, 49.70kg, 57.1kg, 64.56kg,58.35kg,55.21kg, 58.67kg but the weight of one athlete was never recorded. Help determine the missing mass.

    Solution

    Mean = (Sum of all Values)/(Total frequency)

    let the missing value be x

    then total mass for the ten athletes will be 62.10+58.90+56.80+49.70+57.1+64.56+58.35+55.21+58.67+x

    frequency = 10 as there are 10 athletes


    but mean = 60.45

    hence 60.45 =(521.39 + x)/10

    hence 604.5 = 521.39+x

    x=604.45 – 521.39 = 83.06 kg

    Practice Question

    Four numbers have the following number of animals: 134, 233, a, 2a. The mean number of animals owned by the farmers is 250. Find a

    The mode

    Mode is the value that has the greatest occurrence in the data set.

    Consider the following data sets.

    14, 33, 16, 15, 12, 16, 10, 17, 16, 13, 15, 11,9,8,14,16

    16 is repeated most of the times, hence it is the mode.

    Median

    Median is the value that will be in the middle of the data when data is arranged in an ascending order.

    if the total number of items is even number, median is the average of two values in the middle of the data arranged in the ascending order.

    Related Topics