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Tag: Mean

  • 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.

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  • 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.

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