<h5 id="mesastres-of-cental-tendency">Mesastres of Cental Tendency</h5> <ul> <li>Measures of central tendency are statistical techniques used to find the center of distributions.</li> <li>They are also known as statistical averages.</li> <li>The three main measures of central tendency are: <ul> <li>Mean: The arithmetic average of all numbers.</li> <li>Median: The middle number in a sequence of numbers.</li> <li>Mode: The number that appears most frequently in a data set.</li> </ul> </li> <li>The number denoting the central tendency is the representative figure for the entire data set.</li> <li>It is the point about which items have a tendency to cluster.</li> </ul> <p>======</p> <h5 id="mean">Mean</h5> <ul> <li>Mean is a type of average that equally weighs all values in a dataset.</li> <li>For ungrouped data, the individual values are directly summed and divided by the count of the data points.</li> <li>For grouped data, the midpoint of each group (x-bar) is multiplied by the frequency of that group, and the results are summed and divided by the total count.</li> <li>Mean can be calculated using the direct method, where the sum of the values is divided by the count.</li> <li>Mean can also be calculated using the indirect method, where the deviations from the mean are calculated, summed, and divided by the count, then multiplied by -1 and added to the mean’s sum.</li> </ul> <p>======</p> <h5 id="computing-mean-from-ungrouped-data">Computing Mean from Ungrouped Data</h5> <ul> <li>The text compares the Direct and Indirect methods of calculating the mean.</li> <li>The Direct Method involves adding all the observations and dividing by the number of observations.</li> <li>The Indirect Method requires selecting a constant (assumed mean), subtracting it from each observation, and then calculating the mean of the deviations.</li> <li>The formula for the Indirect Method is $\overline{X} = A + \frac{\sum d}{N}$.</li> <li>In the given example, the Direct Method results in a mean of 926.29 mm, which is the same as the mean obtained using the Indirect Method (with 800 as the assumed mean).</li> </ul> <p>======</p> <h5 id="computing-mean-from-grouped-data">Computing Mean from Grouped Data</h5> <ul> <li>The text presents the calculation of the mean for grouped data using both the direct and indirect methods.</li> <li>The direct method involves multiplying each midpoint (X) by its frequency (f) and then dividing the sum of these products by N, the total number of data points.</li> <li>The indirect method uses an assumed mean group with a midpoint (A) to calculate the deviation (d) of each midpoint from A. The mean is then calculated using the formula: mean = A ± ∑fd/N.</li> <li>The text includes a table with data for 99 workers, which is used to demonstrate the calculation of the mean using both methods, resulting in a mean of 102.6.</li> <li>The indirect method can be used with both equal and unequal class intervals.</li> <li>The formula for the sum of fx, ∑fd, and the total number of data points, N, are also provided in the text.</li> </ul> <p>======</p> <h5 id="median">Median</h5> <ul> <li>Median is a positional average.</li> <li>It is defined as the point in a distribution with an equal number of cases on each side of it.</li> <li>The Median is represented by the symbol M.</li> <li>It does not require the data to be in a particular order for calculation.</li> <li>Median is a better measure than mean when data is skewed.</li> </ul> <p>======</p> <h5 id="computing-median-for-ungrouped-data">Computing Median for Ungrouped Data</h5> <ul> <li>The median for ungrouped data is computed by finding the central observation in a arranged series.</li> <li>The central value can be located from either end of the arranged series.</li> <li>The formula used to compute the median is Value of $\left(\frac{\mathrm{N}+1}{2}\right)$ th item.</li> <li>The series is arranged in ascending or descending order and the central value is determined.</li> <li>For example, the median height of mountain peaks in parts of the Himalayas is 8,172 m.</li> </ul> <p>======</p> <h5 id="computing-median-for-grouped-data">Computing Median for Grouped Data</h5> <ul> <li>The text provides a detailed process for calculating the median of a set of data.</li> <li>The steps include setting up a frequency table, calculating cumulative frequencies, determining the median number, identifying the median class, and finally calculating the median using the formula: M = l + i/f(m - c).</li> <li>The median class is the class interval that contains the median number.</li> <li>The median is calculated to be 82.5 in this example.</li> <li>The formula for calculating the median is: M = N/2, where N is the number of values in the data set.</li> </ul> <p>======</p> <h5 id="mode">Mode</h5> <ul> <li>Mode is a measure of central tendency, represented as Z or M0.</li> <li>It is the most frequently occurring value in a distribution.</li> <li>Mode is less commonly used than mean and median.</li> <li>There can be more than one mode in a data set, known as multimodal distribution.</li> <li>Mode is not affected by extreme values or outliers.</li> </ul> <p>======</p>