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Quizzes > High School Quizzes > Mathematics

Standard Deviation Practice Quiz

Enhance your skills with targeted practice problems

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art illustrating trivia for Standard Deviation Showdown aimed at high school students.

What does a small standard deviation indicate about a data set?
Data values are close to the mean
Data values are skewed towards higher numbers
Data contains significant outliers
Data values are widely spread around the mean
A small standard deviation means that the data points are clustered closely around the mean, indicating low variability. This reflects a consistent dataset with little spread among the values.
The standard deviation of a data set is defined as the square root of its:
Variance
Mean
Range
Median
The standard deviation is calculated by taking the square root of the variance. This transformation provides a measure of spread that is in the same units as the original data, making it easier to interpret.
Which symbol is commonly used to represent the population standard deviation?
μ
x̄
s
σ
In statistics, the Greek letter sigma (σ) is used to denote the population standard deviation, while s is typically used for the sample standard deviation. μ and x̄ represent the population mean and sample mean, respectively.
How does the mean absolute deviation differ from the standard deviation?
It squares the differences from the mean
It uses absolute differences instead of squared differences
It divides the sum of differences by the sample size minus one
It is calculated by taking the square root of the average squared differences
Mean absolute deviation calculates the average of the absolute differences from the mean, making it less sensitive to outliers than the standard deviation. In contrast, standard deviation squares the deviations before averaging and then takes the square root.
Given the data set [4, 4, 4, 4], what is the standard deviation?
Undefined
4
2
0
Since all the values in the data set are identical, there is no variability among them, resulting in a standard deviation of 0. This indicates that all data points are equal to the mean.
What is the variance for the following data set: 2, 4, 4, 4, 5, 5, 7, 9?
8
5
4
6
First, the mean of the dataset is 5. Then, computing each squared difference from the mean and averaging them gives a variance of 4. This calculation shows how spread out the values are around the mean.
Which of the following formulas best represents the standard deviation (σ) of a population data set?
√(∑(x - μ)² / N)
(∑(x - μ)²) / N
(∑|x - μ|) / N
√(∑(x - μ)² / (N - 1))
The population standard deviation is calculated by taking the square root of the average squared deviations from the mean, which corresponds to √(∑(x - μ)² / N). The formula using (N - 1) is used for sample standard deviation.
Which of the following is true about the effect of outliers on standard deviation?
Outliers have no effect on the standard deviation
Outliers only affect the mean, not the standard deviation
Outliers decrease the standard deviation
Outliers increase the standard deviation
Outliers cause larger squared differences when computing standard deviation, thereby increasing the measure of spread. This is because the calculation involves squaring the deviations, which amplifies the impact of extreme values.
A class has test scores with a mean of 70 and a standard deviation of 10. Approximately what percentage of scores fall between 60 and 80 in a normal distribution?
34%
95%
50%
68%
In a normal distribution, about 68% of data values lie within one standard deviation of the mean. With a standard deviation of 10, scores between 60 and 80 represent one standard deviation below and above the mean, respectively.
Which example indicates low variability in a dataset?
Daily temperatures vary by 10°C above or below the mean
Temperatures are erratic without any discernible pattern
Daily temperatures vary only by 1°C around the mean
Some days are 15°C, while others are 5°C
Low variability means the data values are very similar and do not deviate much from the average. The option describing a small fluctuation of 1°C around the mean best represents low variability.
To compute the sample standard deviation, which adjustment is made compared to the population standard deviation?
Subtract 1 from the final result
Divide by n-1 instead of n
Divide by n squared
Multiply by n-1
When using a sample to estimate the population variability, the sum of squared deviations is divided by n-1. This adjustment, known as Bessel's correction, accounts for the degrees of freedom in the sample data.
How does standard deviation differ from the range as a measure of variability?
Range provides a more complete picture of data dispersion than standard deviation
They are calculated using the same approach
Standard deviation accounts for every data point rather than just the extremes
Standard deviation only considers the highest and lowest values
Standard deviation involves every data point by measuring how each deviates from the mean, while the range only considers the extreme values (maximum and minimum). This makes standard deviation a more robust measure of overall variability.
In the standard deviation formula, what operation is performed on the differences from the mean before averaging?
Doubling the differences
Taking the cube of the differences
Squaring the differences
Calculating the absolute value of the differences
The formula for standard deviation calls for each difference between the data point and the mean to be squared, ensuring that negative and positive differences contribute equally to the overall variability. Squaring the differences also accentuates larger deviations from the mean.
Which data set has a higher standard deviation?
Data Set A, where values are closer to the mean
Both data sets have the same standard deviation
Data Set B, where values are more spread out
It cannot be determined without more information
A higher standard deviation indicates a greater spread of data values. In this question, Data Set B's values are more spread out from the mean compared to Data Set A, leading to a higher standard deviation for Data Set B.
If the standard deviation of a data set increases, what can be inferred about the set?
The median of the data set decreases
The data values are closer to the mean
The mean of the data set increases
The data values are more spread out from the mean
An increase in standard deviation indicates that the individual data values deviate further from the mean, showing greater dispersion in the dataset. This does not necessarily affect the mean or median, which are measures of central tendency.
Given the data set [3, 5, 7, 8, 10], what is the sample standard deviation (approximate to two decimal places)?
Approximately 2.00
Approximately 3.00
Approximately 2.70
Approximately 1.50
First, the mean of the dataset is calculated as 6.6. Then, using the formula for sample standard deviation which divides the sum of squared differences by (n-1) and taking the square root, the result comes out to be approximately 2.70.
In a large data set, the standard deviation is unusually high. Which of the following is a plausible explanation?
The data are arranged in ascending order
The mean value is exactly zero
The dataset has all identical values
The dataset includes significant outliers that increase overall variability
A high standard deviation often results from the presence of outliers, which are extreme values that drastically increase the variability measure. The other options do not contribute to an increased spread of data.
A researcher adds a constant to every value in a dataset, which increases the mean but leaves the standard deviation unchanged. Why?
Because the constant value cancels out the differences
Because standard deviation measures relative dispersion and is unaffected by a uniform shift
Because standard deviation only considers the extreme values
Because adding a constant decreases the overall variance
Standard deviation quantifies the spread of the data relative to the mean and does not change when all values are shifted by a constant. This is because the differences between the data points remain the same after adding a constant.
If every value in a dataset is doubled, how does the standard deviation change?
It doubles
It remains the same
It is halved
It is squared
Multiplying every data point by a constant multiplies the standard deviation by the absolute value of that constant. Therefore, doubling each value will result in the standard deviation also doubling.
Consider two datasets, A and B, with the same mean. Dataset A's values are tightly clustered around the mean, while Dataset B's values are more spread out. Which of the following is true regarding their standard deviations?
Dataset B has a higher standard deviation than Dataset A
Dataset A has a higher standard deviation than Dataset B
Standard deviation cannot be used to compare the spread in this case
Both datasets have the same standard deviation
Standard deviation measures how much the values in a dataset deviate from the mean. Since Dataset B's values are more spread out compared to Dataset A's, the standard deviation of Dataset B will be higher.
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Study Outcomes

  1. Understand the concept of standard deviation and its importance in measuring data variability.
  2. Calculate the standard deviation for various data sets.
  3. Analyze data distributions to identify patterns and outliers.
  4. Apply statistical formulas to solve real-world problems involving variability.
  5. Interpret the results of data analysis to evaluate the spread of data and its implications.

Standard Deviation Practice Cheat Sheet

  1. Understand what standard deviation measures - Standard deviation shows how spread out numbers are around the mean by calculating the average distance each point deviates. It's like checking how far each friend stands from the group in a photo on average, giving you a sense of cluster tightness. Read more on Math Is Fun
  2. Master the formula differences - The population standard deviation (σ) divides by N, while the sample standard deviation (s) divides by N − 1 to correct for bias when working with a subset of data. Picking the right formula keeps your results accurate and trustworthy. Check the details at Scribbr
  3. Practice the calculation steps - To calculate, first find the mean, subtract each data point from it, square those differences, average the squares, then take the square root. Going through each step manually helps you internalize the process and avoid calculator mix‑ups. Step-by-step guide at Math Is Fun
  4. Link variance and standard deviation - Variance is simply the square of the standard deviation, so it measures dispersion in squared units. While variance tells you how data spread grows quadratically, standard deviation brings it back to the original scale, making interpretation easier. Learn more on BYJU'S
  5. Apply the empirical rule - In a normal distribution, about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. This "68‑95‑99.7" rule is your shortcut to quickly assess how typical or unusual a value is. Discover the empirical rule at Scribbr
  6. Know why samples use N − 1 - Using N − 1 in the sample formula (instead of N) compensates for the fact that samples tend to underestimate variability. This "degrees of freedom" adjustment makes your sample statistics unbiased and more reliable. See the explanation on Math Is Fun
  7. See real-world applications - Standard deviation helps you measure consistency - whether it's test scores in class, daily temperatures, or product quality in manufacturing. Spot large deviations to flag outliers or quality issues before they become big problems. Explore examples on Investopedia
  8. Compare variability across data sets - You can use standard deviation to compare the spread of different data sets even if their means differ. A higher standard deviation signals greater variability, so you'll know which set has more "ups and downs." Compare data at Scribbr
  9. Remember it's always non‑negative - Since standard deviation measures distance, it can never be negative - zero means no spread at all. If you ever calculate a negative value, double‑check your math! Review the concept on GeeksforGeeks
  10. Build confidence with practice sets - Work with diverse data sets - small, large, skewed, or uniform - to reinforce your calculation skills and interpretation. The more you practice, the more intuitive standard deviation becomes, and soon you'll be a stats superstar! Extra exercises at Math Is Fun
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