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Thought-Process to Discover Knowledge

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mathsStatistics and ProbabilityStatistics : Analysis of Data

Central Tendencies of Data : Mean, Median, Mode

In this page, computing mean, median, and mode using tally / table form of data is explained with examples.



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We studied the following example.

The heights of five students from a class are given below in centimeters.

`90`, `97`, `91`, `92`, `90`.

Instead of providing the entire data, the data can be presented in some representative values.

What are the representative values we have studied earlier?

  • Mean, Median, Mode
  • Mean, Median, Mode
  • Flat, Vertical, Horizontal
  • Plane, Surface, Dimension

The answer is "Mean, Median, Mode"

What is mean of data `90`, `97`, `91`, `92`, `90`?finding mean

  • `91`
  • `92`
  • `92`
  • `97`

The answer is "`92`". Add the data values and divide the sum by the number of data-values.

mean
`=(90+97+91+92+90)//5`
`=460//5`
`=92`

What is median of data `90`, `97`, `91`, `92`, `90`?finding median

  • `91`
  • `91`
  • `92`
  • `90`

The answer is "`91`". Arrange the data-values in ascending order and the value in the middle is the median.

median
`=text( middle of increasing order )(90, 97, 91, 92, 90)`
`=text( middle value of )(90, 90, 91, 92, 97)`
`=91`

What is mode of data `90`, `97`, `91`, `92`, `90`?finding mode

  • `91`
  • `92`
  • `90`
  • `90`

The answer is "`90`". Count the number of data-values for each data-value and the most repeated one is the mode.

`90` is repeated twice, and other data points `91`, `92`, `97` are repeated only once. So,
mode
`=90`

Which of the following is a meaning for the phrase "central tendencies"?

  • nature of being or bias of the average/middle
  • nature of being or bias of the average/middle
  • soft and very sensitive in the middle

The answer is "nature of being or bias of the average/middle".

What is the term used to refer "nature of being or bias of the average/middle"?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is "central tendencies".

Mean, Median, and Mode are central tendencies of the data.

Nature of the data is that it has large number of data-values.
Each of the central tendencies represents the data with a representative value.

Consider the data `90`, `97`, `91`, `92`, `90`.

Mean is calculated by adding the data values and dividing the sum by the number of data values.

mean by definition
`=(90+97+91+92+90)//5`
`=460//5`
`=92`

When the data is in tabular form, we have the data value and the count (number of times the data value repeats) of data.

`90` : count `2`
`97` : count `1`
`91` : count `1`
`92` : count `1`
The mean can be calculated as "sum of (value multiplied by count) divided by sum of counts".

mean from tabular form of value-count
`=(90xx2+97xx1+91xx1``+92xx1)//``(2+1+1+1)`
`=460//5`
`=92`

Consider the data: Number of glasses of water students drink during the school time.

The figure illustrates tally and tabular form of data. calculation of mean Calculation of mean is illustrated in the figure. What does this calculation result in?

  • mean of the data is `2.2`
  • mean of the data is `2.2`
  • median of the data is `2`

The answer is "mean of the data is `2.2`"

The figure illustrates finding mean using the tally and tabular form of data.calculation of mean The last column shows, data-value multiplied by its frequency.

In the row marked as "total" the frequency is added and the values in last column are added.

In the row marked as "mean", the sum of data-value multiplied by frequency is divided by the sum of frequency.

Consider the data: Number of glasses of water students drink during the school time.

The figure illustrates tally and tabular form of data. calculation of median Calculation of median is illustrated in the last column. What does this calculation result in?

  • mean of the data is `2.2`
  • median of the data is `2`
  • median of the data is `2`

The answer is "median of the data is `2`"

The figure illustrates finding median using the tally and tabular form of data. calculation of median The last column shows the cumulative frequency.

In the row marked as "total", the frequency is added.

In the row marked as "count in the middle", the total is divided by `2` and the center position is `20` and `21`.

The last column "cumulative frequency" is examined to find the data-value at position `20`. The data value is `2`.

The median value of the given data is `2`.

Consider the data: Number of glasses of water students drink during the school time.

The figure illustrates tally and tabular form of data. calculation of mode Calculation of mode is illustrated in the last column. What does this calculation result in?

  • mode of the data is `2`
  • mode of the data is `2`
  • median of the data is `2`

The answer is "mode of the data is `2`"

The figure illustrates finding mode using the tally and tabular form of data. calculation of mode The last column provides the frequency.

The largest frequency value is selected and highlighted. The corresponding data value is the mode of the data.

The mode of the given data is `2`.

                            
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