Measures of Central Tendency: Mean, Median and Mode

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Get the measures of Central Tendency through Excel for Ungrouped and Grouped data

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Caught in the middle

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How do we describe a set of data as a single number?

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Measures of Central Tendency We often need a single number to represent a set of data. This one number can be thought of as being “typical” of all the data. It intended to describe the center or middle of a set of data

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The Mean is the sum of the values divided by the number of observations (average) The Median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. The Mode is the most frequently occurring value in the data set.

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When do we use mean, median and mode?

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Measurement Scale Best measure of the “middle” Nominal Mode Ordinal Median Interval Symmetrical – Mean Skewed – Median Ratio Symmetrical – Mean Skewed – Median

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MEAN Use the mean to describe the middle of a set of data that does not have an outlier. Advantages Most popular measure in fields such as business, engineering and computer science. It is unique - there is only one answer. Useful when comparing sets of data . Disadvantage Affected by extreme values (outliers )

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MEDIAN Use the median to describe the middle of a set of data that does have an outlier. Advantages Extreme values (outliers) do not affect the median as strongly as they do the mean. Useful when comparing sets of data. It is unique - there is only one answer. Disadvantage Not as popular as mean.

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MODE Use the mode when the data is non-numeric or when asked to choose the most popular item. Advantages Extreme values (outliers) do not affect the mode . Disadvantages: Not as popular as mean and median. Not necessarily unique - may be more than one answer. When no values repeat in the data set, the mode is every value and is useless. When there is more than one mode, it is difficult to interpret and/or compare.

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How do we solve for the mean, median, and the mode?

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Ungrouped or Raw Data are those data which are not yet organized or arranged in a frequency distribution. Grouped Data are those data organized and summarized in the form of a frequency distribution. They have different ways to solve for the mean, median, and mode.

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Ungrouped Data = Mean Sum of all data Number of data

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Highlight the data. To compute for ∑x using Excel Then divide by n.

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Or you can get right away the Mean 83.3

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Highlight the data. Ungrouped Data Median Get the middle. 81.5

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Mode m ost common value bimodal 72 and 100

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Task #1 Measures of Central Tendency (Ungrouped) Save file as T2A1_surname

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How do we solve for the mean, median, and the mode for grouped data?

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Grouped Data = Mean Sum of (frequency * CM) This needs frequency distribution table

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=cell*cell (lower right corner) pointer changes drag to copy the formula of the 1 st cell Get the SUM

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Use the formula to solve for the mean. Mean = 85.6

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Additional Tips for making a Frequency Distribution Table: To identify the HV and LV of a data set: =min(data set) =max(data set) HIGHLIGHT the data set after typing the function

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Instead of using conditional formatting to count the frequency, you may use the formula =COUNTIF(data set,">= 70")- COUNTIF(data set,"> 74") Lower class limit Upper class limit

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Task #2 Measures of Central Tendency (Grouped) Save file as T2A2_surname

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Grouped Data m dn = Lcb me + Median lower class boundary of the median class c umulative frequency below the median class f requency of the median class class width

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Make a Cf < column. Locate the median class. C fb C fb = 21

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Find the L Cb of the median class L Cb = 84.5 fm = ? c = ? f me = 11 c = 5

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Organize the variables Solve for the median m dn = Lcb me +

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m dn = Lcb me +

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Task #3 Measures of Central Tendency (Grouped) Save file as T2A3_surname

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Grouped Data mo = Lcb mo + ( ) c Mode lower class boundary of the modal class d ifference between the frequency of the modal class and the frequency before it d ifference between the frequency of the modal class and the frequency after it class width

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Find the L Cb of the modal class Modal class has the HIGHEST frequency

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Find the D 1 and D 2 D 1 D 2 D 1 = 11-5 = 6 D 2 = 11-9 = 2

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Organize the variables Solve for the mode mo = Lcb mo + ( ) c

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mo = Lcb mo + ( ) c

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Task #4 Measures of Central Tendency (Grouped) Save file as T2A4_surname

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How do the measures of central tendency affect the graphs?

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Negatively Skewed If the mean is less than the median and the median is less than the mode.

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Symmetrical Distribution If the mean, median, and the mode are equal.

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Positively Skewed If the mean is greater than the median and the median is greater than the mode.