Measurements of central tendency
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Measurements of Central Tendency. Statistics vs Parameters. Statistic: A characteristic or measure obtained by using the data values from a sample. Parameter: A characteristic or measure obtained by using all the data values from a population. Notation.

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Statistics vs parameters l.jpg
Statistics vs Parameters

  • Statistic: A characteristic or measure obtained by using the data values from a sample.

  • Parameter: A characteristic or measure obtained by using all the data values from a population.


Notation l.jpg
Notation

  • Roman Numerals: Used to denote statistics (from a sample) X

  • Greek letters: Used to denote parameters (from a total population ( ) (pronounced mu)


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Mean

  • The sum of the values in a sample, divided by the total number of values. The symbol represents the sample mean.

  • The symbol μ is used to represent the mean of a population.


Rounding rule for the mean l.jpg
Rounding rule for the mean

  • The mean should be rounded to one more decimal place than occurs in the raw data.


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Finding the mean from a frequency distribution l.jpg
Finding the mean from a frequency distribution calculating the mean. Sometimes, we are given a frequency distribution and asked to calculate the mean.


Find the mid point of the class l.jpg
Find the mid point of the class calculating the mean. Sometimes, we are given a frequency distribution and asked to calculate the mean.


Multiply the frequency by the midpoint l.jpg
Multiply the frequency by the midpoint calculating the mean. Sometimes, we are given a frequency distribution and asked to calculate the mean.


Sum the last column l.jpg
Sum the last column calculating the mean. Sometimes, we are given a frequency distribution and asked to calculate the mean.



Median l.jpg
Median mean.

  • To find the mean we totaled the values and divided by the number of values.

  • To find the median we arrange the data in order, and select the middle point.


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Example mean.

  • Find the median of 7, 3, 4, 5 , 9

  • Place in order: 3, 4, 5, 7, 9

  • Select the middle point

    5


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Mode the middle two and add them, then divide by 2, to find the median.

  • To find the mean we totaled the values and divided by the number of values.

  • To find the median we arrange the data in order, and select the middle point.

  • To find the mode we find the value that occurs most often in a data set.


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Example the middle two and add them, then divide by 2, to find the median.

  • Find the mode of 3, 2, 4, 6, 7, 2 ,8

  • Since the value 2 occurs twice, and the rest only occur once, the mode is 2


Example18 l.jpg
Example the middle two and add them, then divide by 2, to find the median.

  • Find the mode of 3, 4, 2, 7, 8

  • Each occurs only once, there is no mode

  • Note that the mode is not zero, we say that there is no mode.


Example19 l.jpg
Example the middle two and add them, then divide by 2, to find the median.

  • Find mode of 2, 3, 4, 4, 4, 5, 6, 7,7, 7, 8

  • Observe that both 4 and 7 occur 3 times.

  • We say that the distribution is bi modal, with modes 4 and 7


Midrange l.jpg
Midrange the middle two and add them, then divide by 2, to find the median.

  • To find the mean we totaled the values and divided by the number of values.

  • To find the median we arrange the data in order, and select the middle point.

  • To find the mode we find the value that occurs most often in a data set.

  • To find the midrange, we find the sum of the highest and lowest values in the data set and divide by 2.


Advantages and disadvantages of each method l.jpg
Advantages and Disadvantages of each method the middle two and add them, then divide by 2, to find the median.

Mean

  • Varies less than the median or mode when samples are taken from the same population.

  • Is used for computing other statistics

  • Is unique, not necessarily one of the data values

  • Is affected by extremely low or high values, called outliers.


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Median the middle two and add them, then divide by 2, to find the median.

  • Used when you must find the middle value of a data set

  • Used when you must determine if values fall into the upper half or the lower half of the distribution

  • Affected less than the mean by extremely high or low values.


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Mode the middle two and add them, then divide by 2, to find the median.

  • Used when the most typical case is desired

  • Easiest to compute

  • Used when the data is nominal –political preference, favorite sports team, and the like

  • Not always unique, may not exist


Midrange24 l.jpg
Midrange the middle two and add them, then divide by 2, to find the median.

  • Easy to compute

  • Gives the midpoint

  • Affected by extremely high or low values in a data set.


Right or positive skewed distribution l.jpg
Right or Positive Skewed Distribution the middle two and add them, then divide by 2, to find the median.

Mode Median Mean


Symmetric l.jpg
Symmetric the middle two and add them, then divide by 2, to find the median.


Left or negative skew l.jpg
Left or Negative Skew the middle two and add them, then divide by 2, to find the median.