Normal Distribution — MCQs
APGAR scores of 30 children are recorded in a hospital, and most of the readings are found to be 7 or above. What can be inferred about this data distribution?
In a normal distribution, one standard deviation from the mean includes approximately:
What is the 95% confidence interval for the intraocular pressure (IOP) in the 400 people, given a mean of 25 mm Hg and a standard deviation of 10 mm Hg?
For a positively skewed curve, which measure of central tendency is largest?
What is the shape of a normal distribution curve?
A study is performed to assess the intelligence quotient and the crime rate in a neighborhood. Students at a local high school are given an assessment and their criminal and disciplinary records are reviewed. One of the subjects scores 2 standard deviations over the mean. What percent of students did he score higher than?
Which one of the following tests should be applied to compare mean haemoglobin level of two groups of antenatal mothers?
In a normal distribution, Mean ± 2 S.D. contains
In the estimation of statistical probability, Z score is applicable to:
Which of the following statements is true for a left-skewed distribution?
Normal Distribution Indian Medical PG Practice Questions and MCQs
Question 1: APGAR scores of 30 children are recorded in a hospital, and most of the readings are found to be 7 or above. What can be inferred about this data distribution?
- A. Positively skewed data
- B. Negatively skewed data (Correct Answer)
- C. Normal distribution
- D. Symmetrical data
Explanation: ***Negatively skewed data*** - A distribution is **negatively skewed** when the bulk of the data is concentrated at the **higher end** of the scale - In this case, most **APGAR scores are 7 or above** (out of maximum 10), indicating a **left-skewed or negatively skewed distribution** - The tail of the distribution extends toward the **lower values**, while the peak is at the **higher end** - In negatively skewed data: **Mean < Median < Mode** *Positively skewed data* - **Positively skewed data** would imply that most APGAR scores were at the **lower end** of the scale, with a tail extending toward higher values - This is contrary to the observation that most scores are 7 or above - In positively skewed data: **Mode < Median < Mean** *Normal distribution* - A **normal distribution** implies a **symmetrical bell-shaped curve** where data is evenly distributed around the mean - The description "most readings are 7 or above" clearly indicates an **asymmetrical distribution**, not a normal one - In normal distribution: **Mean = Median = Mode** *Symmetrical data* - **Symmetrical data** means the distribution is balanced, with equal spread on both sides of the center - The given condition that most readings are at the **higher end (7 or above)** signifies an **imbalance**, ruling out symmetry
Question 2: In a normal distribution, one standard deviation from the mean includes approximately:
- A. 50% of the data
- B. 68% of the data (Correct Answer)
- C. 95% of the data
- D. 100% of the data
Explanation: ***68% of the data*** - In a **normal distribution** (bell curve), approximately **68%** of the data falls within **one standard deviation** of the mean. - This is a fundamental property of the **empirical rule** (or 68-95-99.7 rule) for normal distributions. *50% of the data* - **50%** of the data in a normal distribution lies below the **mean**, or within the **interquartile range** if measured from median. - It does not represent the data encompassed by one standard deviation from the mean. *95% of the data* - Approximately **95%** of the data in a normal distribution falls within **two standard deviations** of the mean. - This is another key part of the **empirical rule**, but it refers to a larger range than one standard deviation. *100% of the data* - While theoretically all data points of a continuous distribution are contained somewhere, **100%** of the data is not practically enclosed within a finite number of standard deviations in a true normal distribution. - Virtually all (e.g., 99.7%) of the data falls within **three standard deviations**, but 100% is usually considered to span an infinite range.
Question 3: What is the 95% confidence interval for the intraocular pressure (IOP) in the 400 people, given a mean of 25 mm Hg and a standard deviation of 10 mm Hg?
- A. 22-28
- B. 23-27
- C. 21-29
- D. 24-26 (Correct Answer)
Explanation: ***24-26*** - This is the correct 95% confidence interval calculated using the formula: **mean ± (Z-score × standard error of the mean)**. - For a 95% confidence interval, the **Z-score is 1.96**. - The **standard error of the mean (SEM)** = standard deviation / √(sample size) = 10 / √400 = 10 / 20 = **0.5**. - Therefore: 25 ± (1.96 × 0.5) = 25 ± 0.98 = **24.02 to 25.98**, which rounds to **24-26**. *22-28* - This interval is too wide for a 95% confidence interval with the given parameters. - An interval of ±3 would correspond to a Z-score of 3/0.5 = 6, which is far beyond the **1.96 required for 95% confidence**. - This would represent a much higher confidence level (>99.9%). *23-27* - This interval is slightly too wide, implying a larger margin of error than calculated. - A range of ±2 would require a Z-score of 2/0.5 = 4 times the SEM, which **overestimates the 95% confidence interval**. - This would correspond to approximately 99.99% confidence. *21-29* - This interval is significantly too wide for a 95% confidence interval. - An interval of ±4 would require a Z-score of 4/0.5 = 8 times the SEM, which would correspond to an **extremely high confidence level** (virtually 100%). - This dramatically exceeds what is needed for 95% confidence.
Question 4: For a positively skewed curve, which measure of central tendency is largest?
- A. Mode
- B. All are equal
- C. Median
- D. Mean (Correct Answer)
Explanation: ***Mean*** - In a **positively skewed distribution**, the tail of the distribution extends towards higher values, pulling the **mean** in that direction, making it the largest among the three measures of central tendency. - The presence of **outliers** with large values in the tail disproportionately increases the mean. *Mode* - The **mode** represents the most frequently occurring value in the data set. - In a positively skewed distribution, the mode will be located at the **peak of the distribution**, which is typically the smallest value among the three measures of central tendency. *All are equal* - This statement is characteristic of a **perfectly symmetrical distribution** (e.g., a normal distribution), where the **mean, median, and mode** are all equal. - A positively skewed curve is asymmetrical, meaning these measures will not be equal. *Median* - The **median** is the middle value in an ordered data set, dividing the data into two equal halves. - In a positively skewed distribution, the median will be shifted towards the right of the mode but will still be to the left of the mean, meaning it is **smaller than the mean**.
Question 5: What is the shape of a normal distribution curve?
- A. J-shaped
- B. U-shaped
- C. Bell-shaped (Correct Answer)
- D. None of the options
Explanation: ***Bell-shaped*** - A **normal distribution** is a **symmetric probability distribution** centered around its mean, with tails that taper off indefinitely. - The distinctive shape resembles a **bell**, with the highest point at the mean and gradually decreasing frequencies as values move away from the mean. *J-shaped* - A **J-shaped curve** typically describes a distribution where the frequency is highest at one end and then continuously decreases or increases to the other end. - This shape is not characteristic of the **symmetry** and **central tendency** observed in a normal distribution. *U-shaped* - A **U-shaped curve** indicates that frequencies are highest at both ends of the distribution and lowest in the middle. - This is the opposite of a **normal distribution**, where the highest frequency is at the center (mean). *None of the options* - The term **bell-shaped** accurately describes a normal distribution curve, making this option incorrect.
Question 6: A study is performed to assess the intelligence quotient and the crime rate in a neighborhood. Students at a local high school are given an assessment and their criminal and disciplinary records are reviewed. One of the subjects scores 2 standard deviations over the mean. What percent of students did he score higher than?
- A. 95%
- B. 97.5% (Correct Answer)
- C. 68%
- D. 99.7%
Explanation: ***97.5%*** - This question relates to the **normal distribution (bell curve) and the empirical rule (68-95-99.7 rule)** [1]. - A score 2 standard deviations above the mean means that 95% of the data falls within +/- 2 standard deviations of the mean [2]. This leaves 5% outside of this range (2.5% on each tail). Therefore, the student scored higher than 95% + 2.5% = **97.5%** of students. *95%* - This percentage represents the data that falls within **2 standard deviations of the mean (both sides)**, not the percentage a score 2 standard deviations above the mean is higher than [1]. - It would be correct if the question asked for the percentage of students whose scores fall within two standard deviations of the mean. *68%* - This percentage represents the data that falls within **1 standard deviation of the mean** according to the empirical rule [1]. - A score 2 standard deviations above the mean is significantly higher than this range. *99.7%* - This percentage represents the data that falls within **3 standard deviations of the mean (both sides)**, according to the empirical rule [2]. - This would mean the student scored 3 standard deviations above the mean, which is not stated in the question.
Question 7: Which one of the following tests should be applied to compare mean haemoglobin level of two groups of antenatal mothers?
- A. Analysis of variance
- B. Chi-square test
- C. Unpaired t-test (Correct Answer)
- D. Paired t-test
Explanation: ***Unpaired t-Test*** - The **unpaired t-test** is used to compare the means of **two independent groups** on a continuous variable, such as hemoglobin levels. - Antenatal mothers in two distinct groups are independent, and **hemoglobin level is a continuous variable**, making this the appropriate choice. *Analysis of variance* - **ANOVA** (Analysis of Variance) is used to compare the means of **three or more independent groups**. - Since there are only **two groups** being compared, ANOVA is not the most efficient or appropriate test. *Chi-square test* - The **Chi-square test** is used to analyze the association between **two categorical variables**. - Hemoglobin level is a **continuous variable**, not categorical, so this test is not suitable for comparing means. *Paired t-test* - The **paired t-test** is used to compare the means of **two related groups** or the same group measured at two different times (e.g., before and after an intervention). - The two groups of antenatal mothers are **independent**, not paired or related.
Question 8: In a normal distribution, Mean ± 2 S.D. contains
- A. 68.3 % values
- B. 95.4 % values (Correct Answer)
- C. 91.2 % values
- D. 99.7 % values
Explanation: ***95.4 % values*** - According to the **empirical rule** (or 68-95-99.7 rule) for normal distributions, approximately **95.4%** of data falls within two standard deviations of the mean. - This interval covers from (Mean - 2 S.D.) to (Mean + 2 S.D.) and represents the likelihood of a value falling in this range. *68.3 % values* - This percentage corresponds to the data contained within **Mean ± 1 S.D.** in a normal distribution, not Mean ± 2 S.D. - It signifies that roughly two-thirds of all observations lie within one standard deviation from the mean in a bell-shaped curve. *91.2 % values* - This value is not a standard percentage associated with common multiples of standard deviations (1, 2, or 3) from the mean in a normal distribution. - It does not correspond to any universally recognized interval like ±1 S.D., ±2 S.D., or ±3 S.D. *99.7 % values* - This percentage represents the data contained within **Mean ± 3 S.D.** in a normal distribution. - It indicates that almost all (99.7%) of the data points are expected to fall within three standard deviations from the mean.
Question 9: In the estimation of statistical probability, Z score is applicable to:
- A. Poisson distribution
- B. Normal distribution (Correct Answer)
- C. Skewed distribution
- D. Binomial distribution
Explanation: ***Normal distribution*** - The **Z-score** (or standard score) is a measure of how many **standard deviations** an element is from the mean. It is specifically used when working with **normally distributed data**. - It allows for the comparison of scores from different normal distributions by standardizing them to a common scale. *Poisson distribution* - This distribution deals with the **number of events** occurring in a fixed interval of time or space, given a known average rate, and is not typically used with Z-scores directly. - It is a **discrete probability distribution**, unlike the continuous nature required for direct Z-score application. *Skewed distribution* - A skewed distribution has an **asymmetrical shape**, where points cluster more on one side of the mean. - Z-scores can be calculated for skewed distributions, but their interpretation as probabilities (e.g., using a standard normal table) is **not valid** because the data do not follow a bell-shaped curve. *Binomial distribution* - This distribution describes the **number of successes** in a fixed number of independent Bernoulli trials. - It is a **discrete probability distribution** and generally, Z-scores are not directly applied to it, although for a large number of trials, it can be approximated by a normal distribution.
Question 10: Which of the following statements is true for a left-skewed distribution?
- A. Mean = Median
- B. Mean>Mode
- C. Median > Mean (Correct Answer)
- D. Mean < Mode
Explanation: ***Median > Mean*** - In a **left-skewed distribution**, the bulk of the data is on the right, and the tail extends to the left, pulling the **mean** towards the lower values. - This pull results in the **mean** being less than the **median**, which is less affected by extreme values in the tail. *Mean = Median* - This relationship holds true for a **symmetrical distribution**, such as a **normal distribution**, where the data is evenly distributed around the center. - In a **skewed distribution**, the mean and median will diverge due to the presence of outliers or extreme values on one side. *Mean>Mode* - This statement is characteristic of a **right-skewed distribution**, where the tail extends to the right, pulling the **mean** to a higher value than the **mode**. - In a right-skewed distribution, typically **mode < median < mean**. *Mean < Mode* - This statement indicates that the **mode** (the most frequent value) is greater than the **mean**, which is not a defining characteristic of a left-skewed distribution. - While it can occur, the primary relationship for left-skewness is **mean < median**.
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