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Statistical Measures

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Statistical Measures

Statistical measures are a descriptive analysis technique used to summarise the characteristics of a data set. This data set can represent the whole population or a sample of it. Statistical measures can be classified as measures of central tendency and measures of spread.

What are measures of central tendency?

Measures of Central Tendency describe some key characteristics of the data set based on the average or middle values, as they describe the centre of the data. The measures of central tendency that we will be looking at are the mean, mode, and median.

Mean

The mean, also called the mathematical average of a given data set, can be found by adding all values in the data set, and dividing by the number of values. We can use a mathematical formula to describe this : where 𝜇 is used to represent the mean.

We have the scores of a quiz taken by mathematics students in the grade. They are 76, 89, 45, 50, 88, 67, 75, 83. What is the mean score?

Answer:

The formula above means we will add all the scores and then divide the sum by the number of scores available.

76 + 89 + 45 + 50 + 88 + 67 + 75 + 83 = 573

Since there are 8 scores available, we will divide our sum by 8.

𝜇 = 71.625

Mode

The mode is the most frequently occurring value in a data set. Sometimes you will have a data set where this describes more than one value. Here they are all considered the mode.

Find the mode for the given data set 6, 9, 3, 6, 6, 5, 2, 3.

Answer:

Arranging these values in ascending order will help you to identify which one occurs the most.

2, 3, 3, 5, 6, 6, 6, 9

It is evident that 6 is the most frequently occurring number, therefore the mode is 6.

Median

The median is the midpoint value of a given data set. In cases where the midpoint values are two (when the number of data points is even), you need to find the average of both middle values. When finding the median, it is appropriate to reorder your values in ascending order. Take the value if the number of data points is odd. When the number is even, take the and the value.

The ages of 12 students in grade 11 were collected, and the values are as follows: 15, 21, 19, 19, 20, 18, 17, 16, 17, 18, 19, 18. Find the median age.

Answer:

Arrange these values in ascending order:

15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 21

Since the number of data points is even, we will have two middle numbers, which are both 18. So the median is 18.

The scores of an exam taken by 7 students are given below. Find the median score.

87, 56, 78, 66, 73, 71, 79

Answer:

Rearrange the numbers from lowest to highest.

56, 66, 71, 73, 78, 79, 87

The number of value points is odd, so the middle number becomes the median score.

Median = 73

What are measures of spread?

Measures of spread are statistical measures that describe the similarity and variety of the values of given datasets. Relying on central tendency measures alone as a summary description for data sets can be very misleading since it does not account for extreme values. Measures of spread help us do that, including range, variance, and standard deviation.

Range

The range is the difference between a given data set's highest and lowest values. It helps you to know how wide the data is. To find the range, the lowest value in the data is subtracted from the highest value.

Find the range of the ages of 12 students in a class. Here's your data: 15, 21, 19, 19, 20, 18, 17, 16, 17, 18, 19, 18.

Answer:

Highest value = 21

Lowest value = 15

Range = highest value - lowest value

Range = 21-15

Range = 6

However, the range has a few limitations:

  • It is affected by outliers.

  • It cannot be used for open-ended distribution.

Quartiles and the interquartile range

A quartile is a type of quantile that divides an ordered data set into four parts (quarters). A quartile is not the group of numbers that have been divided. It is the cut-off point in the division.

The interquartile range is the difference between the upper quartile and the lower quartile value.

To find the quartile of a given data set you can proceed as follows:

  1. Order the values in ascending order.

  2. Find the median. This is always labeled as the second quartile ( ).

  3. Now find the median of both halves of the data set. The lowest half is labelled , and the highest half is labelled .

  4. Find the interquartile range (IQR) by subtracting Q1 from Q3.

Find the interquartile range for the data given 6, 9, 3, 6, 6, 5, 2, 3, 8.

Answer:

  1. Reorder the values from lowest to highest.

2, 3, 3, 5, 6, 6, 6, 8, 9

  1. Find the median

The median is 6.

= 6

  1. Find the median of the two halves, which are: 2, 3, 3, 5 | 6, 6, 8, 9

For the first part, we have 3 as the median.

With the second step, we will have to sum both middle values and divide them by 2.

  1. Find the interquartile range.

IQR = 7-3

IQR = 4

Variance and standard deviation

Variance and standard deviation are both measures of variability. The variance is the measure of how data points vary from the mean, and the standard deviation is the square root of variance. What this tells us is that standard deviation is derived from variance.

Variance is denoted by

Standard deviation is denoted by 𝝈.

Variance formula

The population variance formula is

Where = population variance

N = size of the population

= each value from the population

𝜇 = the population mean.

The sample variance formula is

Where = sample variance

n = size of sample

= each value from the sample

= the sample mean.

Standard deviation formula

The population standard deviation formula is given by

Where 𝝈 = population standard deviation.

N = size of the population.

= each value from the population.

𝜇 = the population mean.

The sample standard deviation formula is given by

Where s = sample standard deviation.

n = size of sample.

= each value from the sample.

= the sample mean.

Calculate the standard deviation for the following scores on a Maths exam taken by -grade students: 82, 93, 98, 89, 88.

Answer:

The first thing you need to do is to find the mean of the sample:

So the formula that we are going to use here is , since the scores are available are only a sample of the whole population of students that took the exam.

We can construct a table to break down the formula and work it out appropriately.

82

64

93

3

9

98

64

89

-1

1

88

-2

According to the formula we will have to sum , which is the last column of our table.

s = 5,958

Standard deviation is 5.958

By definition, variance should be

Statistical Measures - Key takeaways

  • Statistical Measures are a technique of descriptive analysis used to give a summary of the characteristics of a data set.
  • Measures of central tendency describe some key characteristics of the data set based on the average or middle values, as they describe the centre of the data.
  • The three main measures of tendency are mean, mode, and median.
  • Mean is the most common measure of central tendency and its formula is .
  • Measures of spread are statistical measures that describe the similarity and variety of values of given datasets.
  • Standard deviation is a measure of the amount of variation or dispersion of a set of values.
  • Standard deviation is the square root of the variance.

Frequently Asked Questions about Statistical Measures

First, examine components of the data to see if there are any patterns where you can draw conclusions. Now you can explain what these findings mean in context.

Frequency distribution, standard deviation and mean.

 A smaller standard deviation means greater consistency.

 The extent to which two variables are linearly related.

Yes, discrete data is a type of numerical data and measures of central tendency such as mean are commonly used to analyse them.

Final Statistical Measures Quiz

Question

What are statistical measures?

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Answer

Statistical measures are a technique of descriptive analysis used to give a summary of the characteristics of a data set.


Show question

Question

Which of these measures of central tendency best describe the most frequently occurring number in a dataset?


Show answer

Answer

Mode

Show question

Question

What are the three main measures of central tendency?


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Answer

Mean, median, and mode.

Show question

Question

What is the median?


Show answer

Answer

The median is the mid-point value of a given dataset.

Show question

Question

Given the data set {2, 3, 4, 6, 7, 7, 8, 9}, what is the median here?


Show answer

Answer

6.5

Show question

Question

What is the mode if you are given the dataset {2, 5, 3, 2, 5, 6, 7, 5}?


Show answer

Answer

5

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Question

Which of these is not a measure of spread?


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Answer

Mean

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Question

Which of these statements is true about variance and standard deviation?


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Answer

Standard deviation is the square root of variance.


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Question

The difference between the highest values and lowest values of a given data is known as?


Show answer

Answer

The range

Show question

Question

Find the range for the given dataset, {43, 34, 78, 16}.


Show answer

Answer

62

Show question

Question

The difference between the upper quartile and the lower quartile value is known as?


Show answer

Answer

The interquartile range.

Show question

Question

Find the mean of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

14

Show question

Question

Find the mode of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

5

Show question

Question

Find the median of the following set of data: 1,2,2,3,3,5,5,5,6,9,113

Show answer

Answer

5

Show question

Question

When can mean not be used?

Show answer

Answer

When the data is non-numerical

Show question

Question

Find the mean of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2


Show answer

Answer

5.225

Show question

Question

Find the mode of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2

Show answer

Answer

7

Show question

Question

Find the median of the following data, which is shoe size in a class.

Shoe Size         Frequency

2                         4

3                         4

4                         5

5                         7

6                         8

7                         10

8                         2

Show answer

Answer

5.5

Show question

Question

Find the mean of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

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Answer

66.1cm

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Question

Find the mode of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

Show answer

Answer

14cm

Show question

Question

Find the median of the following lengths: 12cm, 14cm, 14cm, 56cm, 76cm, 1.24m, 167cm

Show answer

Answer

56cm

Show question

Question

What is standard deviation?

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Answer

Standard deviation is a measure of dispersion used in statistics to tell us how spread out the values are from the mean in a data set.

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Question

What is the formula for standard deviation in words?

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Answer

The square root of the sum of how far each data point is from the mean squared divided by the total number of data points.

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Question

 What do the symbols in the standard deviation formula stand for?

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Answer

𝝈 is the standard deviation, ∑ is the sum, xi is an individual number in the data set,𝜇 is the mean of the data set and N is the total number of values in the data set.

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Question

What type of distribution do data sets for which we can do a standard deviation follow?  

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Answer

A normal distribution.

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Question

What percentage of data points fall within ±1𝝈 of the mean?

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Answer

68.2%

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Question

What is the standard deviation for the following data set: 5, 10, 13, 14, 18, 19, 21?

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Answer

5.175

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Question

What is the standard deviation for the following data set: 132, 189, 176, 165, 154, 133, 178?

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Answer

20.66

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Question

What is the standard deviation for the following data set: 1.25, 1.26, 1.24, 1.27, 1.27, 1.28?

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Answer

0.0134

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Question

What is the mean of the following data set: 13, 16, 24, 12, 11, 23, 22, 20?

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Answer

17.625

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Question

What is the mean of the following data set: 223, 230, 279, 365, 199, 254?

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Answer

258.33

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Question

In the following data set: (998, 880, 1001, 956, 937, 892), 68.2% of data points will fall between which two values?

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Answer

897.17 and 990.83

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Question

In the following data set: (1.50, 1.40, 1.35, 1.64, 1.39, 1.52, 1.44) 68.2% of data points will fall between which two values?

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Answer

1.37 and 1.55

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Question

What percentage of values fall between 2 standard deviations of the mean?

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Answer

91.4%

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Question

In the following data set: (45, 46, 39, 50, 55, 43, 38, 51), 68.2% of data points will fall between which two values?  

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Answer

40.35 and 51.41

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