Summarizing Data with Numerical Values Introduction: to summarize a set of numerical data we used three types of groups can be used to give an idea about data, these groups are: 1. Central Tendency (النزعة المركزية) such : Mean – Median – Mode. 2. Variation measures (مقاييس التشتت) such as: Range- Mean deviation- variance- standard deviation- coefficient of variation- inter quartile range. 3. Shape such as: Skewness (الالتواء) - Kurtosis (التفلطح).

Central Tendency Mean –Arithmetic mean (Average) Definition: The arithmetic mean of a set of values is the measure of center by adding the values and dividing the total by the number of values. The arithmetic mean is the most widely used measure of location and it can be divided to: Mean of ungrouped data and it can be calculated as:

Central Tendency For sample For population Example: Find the mean of {5.4, 1.1, 0.42, 0.73, 0.48, and 1.1}.

Central Tendency Example: Find the mean of {5.4, 1.1, 0.42, 0.73, 0.48, and 1.1}.

Central Tendency 1.Weighted mean: It is similar to an ordinary arithmic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others.

Central Tendency 1.Weighted mean: Example: Grades are often computed as a weighted average suppose that homework count 10%, quizzes 20% and final 70%. If the student x has a homework 92, a quiz 68 and final 81. Find the weighted mean.

Central Tendency Weighted mean:

Central Tendency Mean from a frequency distribution or grouped data Suppose we have the frequency distribution table as: levelFrequency (f)class midpoint (x)f.x Total

Central Tendency Then Mean is

Central Tendency Properties of the arithmetic mean 1. Every set of interval level or ratio level data has a mean. 2. All the values are included in computing the mean. 3. A set of data has a unique mean.

Central Tendency Properties of the arithmetic mean 4.The mean is affected by unusually large or small data values (outliers). 5. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero.

Central Tendency Median Definition: The median of a data set is the measure of center that is the middle value when the original data value are arranged in order of increasing magnitude the median is often denoted by [ pronounced (x tilde).

Central Tendency Median of ungrouped data To find the median, first sort values from smallest to largest, the follow one of these procedures: If the number of values is odd, then the median is the number located in exact middle of the list. If the number of values is even, then the median is found by computing the mean of two middle numbers

Central Tendency Median of ungrouped data Example 1: suppose we have a measurement: {5.4, 1.1, 0.42, 0.73, 0.48, 1.1,.66} Solution: first arrange the values in order as {0.42, 0.48, 0.66, 0.73, 1.1, 1.1, 5.4} Because the number of values is an odd (7), then the median is the value in the exact middle of sorted list, here is 0.73.

Central Tendency Median of ungrouped data Example 2: suppose we have the observations {7, 4, 3, 5, 6, 8, 10, 1}, find the median of this data set. –Solution: first we sort the data set ascending as: {1, 3, 4, 5, 6, 7, 8 10} –Because the number of values is an even (8), then the median is found by computing the mean of the two middle value of (5,6) as

Central Tendency Median of grouped data or in a frequency distribution table –In a grouped distribution, the following steps should be followed: –Step (1): Form the cumulative frequency (F). –Step (2): find the value of N/2 Where N is sum of frequency –Step (3): Find F value the first exceed N/2 which identified the median class M

Central Tendency Median of grouped data or in a frequency distribution table Step (4): calculate the median using the following formula

Central Tendency Where: The lower bound of the median class The cumulative frequency of class immediately prior to the median class. The actual frequency of median class. The median class width

Central Tendency Median Example: 1- Formulate cumulative frequency AgeFrequencyCumulative frequency Total130

Central Tendency Median Step (2): N/2 = 130/2 =65 Step (3): Median class is [35-40] Step (4):

Central Tendency Median

Central Tendency Median Properties of the median 1. It can be computed for ratio level, interval and ordinal level data. 2. There is a unique median for each data set. 3. It is not affected by extremely large or small values; therefore it is an available measure of central tendency when such values occur.

Central Tendency Median Advantages of median over the mean 1. It may be determined even if the values of all observations are not known (3, 4, 5, x1, x2). 2. Extreme values in data set do not affected on median as strongly as they do on mean.