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Tag: assumed mean

  • 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

  • Working with assumed mean

    Working with assumed mean

    Assumed mean is a certain value that is chosen from the data set such that it can be subtracted from all other values to reduce the size of numbers in the data set.

    An assumed mean is usually determined by guessing the number that could be used as the mean among the values in the data set.

    It is like picking one of the numbers in the dataset and assuming it is the mean for the data. By looking at the data, we can guess a number close to the mean because mean, as a measure of central tendency, which most likely will be a number near the median of the data.

    Take for instance the data set below.

    89, 64, 56, 78, 88, 67, 72, 85, 70, 65, 64, 66, 72, 74, 76.

    Arranging the data in ascending order we have

    56, 64, 64, 65, 66, 67, 70, 72, 72, 74, 76, 78, 85, 88, 89.

    Range= 89 – 56 =33

    A method I find convenient to find a central data item is 56+(33/2) =56+17=73.

    Now because we don’t have 73, I pick 72 as the assumed mean. And I will subtract 72 from each data item as in table below.

    Now i get the summation of fd: ∑fd=-1

    and mean of d= (∑fd)/(∑f) ≈ -0.06667

    mean of x, x̄ = 72 + (-0.0667) = 71.9333

    The sum of x has been done by a statistical software. Otherwise it could be time consuming and error prone and energy sucking to try and compute it manually. It has been recorded there for comparison purposes.

    in a more general case, if we take an assumed mean A and subtract it from each data item x, then we get data items labelled with d such that d=x-A and the mean for the data expressed as:

    Practice question

    Using an assumed mean, calculate the mean of each of the following sets of data

    (a)

    0.655, 0.685, 0.705, 0.665, 0.695, 0.715, 0.375, 0.745, 0.755, 0.765, 0.745, 0.550, 0.450, 0.400, 0.425, 0.325, 0.775, 0.685, 0.695, 0.745

    (b)

    225, 400, 300, 225, , 525, 325, 600, 225, 325, 400, 525, 575, 625, 250, 650, 350, 475, 550, 575, 275, 375, 475, 475, 675, 400, 375, 275, 250.

    Related Topics