Statistics Measures of Centrality and Spread

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Questions and Answers

What is the mode in the discrete data set of idle days: 2, 6, 2, 4, 1, 4, 3, 1, 1?

1 (correct)

Which measure of centrality represents the middle value of a data set when ordered?

Median (correct)

Mean

Mode

Range

In the calculation of the mean, what does the symbol $ ext{∑}$ represent?

Average value

Frequency of values

The sum of all values (correct)

The number of data points

What type of data is characterized by individual values such as people or cars?

Discrete data Signup and view all the answers

What is the mean of the number of idle days based on the grouped data for 95 cars?

2.30 Signup and view all the answers

Which of the following correctly describes continuous data?

Ranges rather than distinct values Signup and view all the answers

What is the median of the discrete data set: 1, 1, 1, 2, 2, 3, 4, 4, 6?

2 Signup and view all the answers

How is the mean calculated for grouped data?

Sum of the product of frequencies and values / Total frequency Signup and view all the answers

What is the inter-quartile range of idle days for the 95 staff cars?

2 days Signup and view all the answers

What is the semi-inter-quartile range for the idle days of the staff cars?

1 day Signup and view all the answers

What is the standard deviation for the sample of 95 staff cars?

1.345 Signup and view all the answers

If the range of idle days is from 0 to 6, what is the total range?

6 days Signup and view all the answers

What is the mid-value of the value range for users of £63 but less than £64?

63.5 Signup and view all the answers

How many users fall into the range of £62 but less than £63?

6 users Signup and view all the answers

What is the total count of car users in the provided data for the fuel invoices?

34 users Signup and view all the answers

What is the value of 'fx' for the mid-value of £60.5 per 5 users?

302.5 Signup and view all the answers

What is the modal class of car values based on the provided data?

£64 but < £65 Signup and view all the answers

What is the mean value of the car prices based on the data?

£62.97 Signup and view all the answers

How is the standard deviation for a population calculated?

By taking the square root of the variance. Signup and view all the answers

What does the semi-inter-quartile range represent?

Half the range of the middle half of the ordered data. Signup and view all the answers

In the example given, what is the range of idle days for the nine members of staff?

5 days Signup and view all the answers

What represents the frequency in the context of standard deviation calculation?

The number of occurrences of each specific data value. Signup and view all the answers

What is the inter-quartile range for the example of idle days provided?

3 days Signup and view all the answers

Which statistical measure reflects the average spread of the data around the mean?

Standard deviation Signup and view all the answers

When using sample standard deviation, what adjustment is made to the sample size in the denominator?

The sample size is decreased by one. Signup and view all the answers

Which measure of spread would be appropriate if all data points are known?

Population standard deviation Signup and view all the answers

What is the semi-inter-quartile range for the given data?

£1 Signup and view all the answers

Which car value has the highest number of users?

64.5 Signup and view all the answers

What is the total number of users in the sample?

34 Signup and view all the answers

What does the histogram indicate about the frequency of car mileages in the range of 120 to 130 miles?

There were 10 cars in that mileage range Signup and view all the answers

From the summarized statistics, which value is used as a population estimator?

£1.958 Signup and view all the answers

How is the mode estimated using the histogram?

By finding the highest frequency column Signup and view all the answers

Which range of car values includes a total of 4 users?

61 but < 62 Signup and view all the answers

What is the estimated mode of car mileages according to the histogram?

146 miles Signup and view all the answers

What does the inter-quartile range indicate in a data set?

The difference between the upper quartile and the lower quartile Signup and view all the answers

Which statistic should be used for ordinal data to represent variability?

Inter-quartile range Signup and view all the answers

If the median value is $146$ miles, what does this indicate about the data set?

50% of the data values are below 146 miles Signup and view all the answers

What is the upper quartile value in the given data set?

156 miles Signup and view all the answers

Which measure is most appropriate for representing the central tendency of interval data?

Mean Signup and view all the answers

Study Notes

Numerical Data Summary

Data can be summarised by two parameters: measures of centrality and measures of spread.
Measures of Centrality indicate a typical or central value.
- Mode: The most frequently occurring value in a dataset.
- Median: The middle value of a dataset ordered from smallest to largest.
- Mean: The arithmetic average of all values in a dataset.
Measures of Spread show how widely the data is distributed around its center.
- Range: The difference between the largest and smallest values.
- Inter-quartile Range (IQR): The range of the middle 50% of the data.
- Semi-inter-quartile Range (SIQR): Half the value of the IQR.
- Standard Deviation: A measure of the average deviation of data points from the mean.
Standard Deviation can be calculated for both populations and samples.
- Population Standard Deviation (σ): A measure of the spread of data in an entire population.
- Sample Standard Deviation (s): An estimate of the population standard deviation based on a sample of data.
The choice of the most appropriate measure of centrality and spread depends on the type of data.
- For ordinal data, use the median and IQR.
- For interval or ratio data, use the mean and standard deviation.
Graphical representations of data can be used to estimate summary statistics.
- Histograms can be used to estimate the mode.
- Cumulative frequency diagrams (ogives) can be used to estimate the median and interquartile range.

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Description

This quiz covers the key concepts of measures of centrality and spread used in summarizing numerical data. You'll explore the mode, median, and mean, along with range, inter-quartile range, and standard deviation. Test your understanding of these essential statistical measures and their applications.