"How many of the 10 running times are more than one SD below the mean" means how many data points from given 10 are less than mean-1SD.

We are given that SD=22.4, so we should find mean --> mean=100 --> there are only 2 data points below 100-22.4=77.6, namely 70 and 75.

Answer: B.

Similar problems:
math-questions-mean-score-88502.html?hilit=below%20mean%20deviation
statistics-question-99221.html?hilit=below%20mean%20deviation#p764887

Hope it's clear.
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"How many of the 10 running times are more than one SD below the mean" means how many data points from given 10 are less than mean-1SD.

Bunuel, I don't get this - It says how many are more than mean-1SD (one SD below the mean)? Why do you say less? Please explain. Thanks.

How many of the 10 running times are more than one SD below the mean.
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How many of the 10 running times are more than one SD below the mean.

So then 1SD below the mean is 77.6. It asks how many of the 10 running times are more than 77.6? I just replaced the one SD below the mean by the value. There are more than 2... What am I missing?

You have a problem with wording.

WRONG reading:
How many of the 10 running times are more than one SD below the mean. So more than Mean-SD.

CORRECT reading:
How many of the 10 running times are more than one SD below the mean. So less than Mean-SD.
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70 75 80 85 90 105 105 130 130 130

The list shown consist of the times, in seconds, that it took each of 10 school children to run a distance of 400 meter. If the SD of ten running times is 22.4 seconds, rounded to nearest tenth of second, how many of the 10 running times are more than one SD below the mean of the 10 running times?
A. one
B. two
C. three
D. four
E. five

The most time consuming part in this question is to define the mean. Under exam pressure and time pressure it is very easy to make mistake.
it is easier to group numbers: 130*3=390; 105*2=210; 75+85=160; 70+80=150; 90;
Next stage combine results, again using more convenient ways to calculate: 390+210=600; 160+150=310; 90. 600+310+90=1000. Since there are 10 numbers the mean is 100. Questions asks to find the quantity of numbers one SD BELOW the mean, which is 100-22,4=77,6. There are only two numbers below 77,6. The answer is B

Pardon me for my understanding, but "are <more than> <one SD below the mean> of the 10 running times?
So isn't it asking values greater than 77.6 ?

WRONG reading:
How many of the 10 running times are more than one SD below the mean. So more than Mean-SD.

CORRECT reading:
How many of the 10 running times are more than one SD below the mean. So less than Mean-SD.
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Let’s calculate the average:

(70 + 75 + 80 + 85 + 90 + 105 + 105 + 130 + 130 + 130)/10 = 1000/10 = 100

One standard deviation below the mean is 100 - 22.4 = 77.6, so there are 2 running times more than 1 standard deviation below the mean.

Answer: B
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The average (mean) is:

(70 + 75 + 80 + 85 + 90 + 105 + 105 + 130 +130 +130)/10 = 1000/10 = 100

Thus, 1 standard deviation below the mean is 100 - 22.4 = 77.6, so there are 2 running times that are more than 1 standard deviation below the mean.

Answer: B
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I am still not understanding the one SD less than mean logic. Why should it be less and not more than 77.6? VeritasKarishma can you help me out, please?

harshbirajdar

Mean =100 (on calculation)

What is the meaning of 1 SD below mean? This is what it means graphically:



For a normal distribution (usually this is what we deal with), 68% values lie within 1 SD of mean (34% below mean and 34% above mean)
95% values lie within 2 SD etc.

So if mean of a normal distribution is say 100 and SD is say 25, there will be 34% values lying within 75 to 100 and 34% values within 100 to 125.

The data might look something like this: ... 50, 76, 78, 93, 102, 115, 120, 134 ...

3 values lying in 1 SD below mean and 3 values lying in 1 SD above mean.

In a question like this (we don't have to worry whether it is a normal distribution), we can find the mean and we are given the SD.
1 SD below mean would be values lying between 77.6 and 100.
"more than one SD below the mean" would be values lying below 77.6 (they are lying within 2 SD or 3 SD or 4 SD...)
There are only 2 such values.
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Hi VeritasKarishma - A quick follow up on this chart specifically.

The question is asking about how many values are there in set below the Mean - 1 standard deviation threshold

Per the graph, there should be only 16 % of the values in any normal distribution { (100 - 68)/2 }

Thus i thought only 16 % of the value in the set

Given the set is 10 elements, 16 % of 10 is 1.6 values

Thus i thought, well we cant have 1.6 values, so i chose 1 value because i thought there will certainly be only one value. 2 values is above 1.6

Why doesn't this logic work

Thank you for everything !

Your logic is correct for a 'normal distribution' i.e. when the data looks like what it is in the graph. This is not a normal distribution since the data does not follow the graph.
When they want you to assume 68% as 1 SD etc, they will tell you that the data is 'normally distributed'.
I gave the graph to show you the concept of 1 SD below mean, 2 SD below mean etc.
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Let's first find the mean.

Mean = \(\frac{Sum }{ Total }\)

=> Mean = \(\frac{70 + 75 + 80 + 85 + 90 + 105 + 105 + 130 + 130 + 130 }{ 10}\)

=> Mean = \(\frac{1000 }{ 10}\)

=> Mean = 100.

Given: S.D. = 22.4

Find: 1 standard deviation below the mean => Mean - 1S.D.

=> 100 - 22.4 = 77.6

Two running times (70 and 75) are below 77.6.

Answer B
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I still dont understand the concept here. The question can be interpreted in 2 different ways:
1. "(values more than) one SD below the mean". This implies how many values are more than 1 SD below the mean which is equal to 100-22.4= 77.6. This would imply all values more than 77.6 which gives us 5 as the answer.
2 . "values (more than one) SD below the mean". In this scenario, the "more than one" acts as an adjective for SD . In this case, I understand that we are looking for 2sigma, 3 sigma values below one SD below the mean. This would give us the answer as 2. VeritasKarishma

"more than 1 standard deviation below the mean"

means

"more than 22.4 below the mean"

means

"22.5 below the mean" or "23 below the mean" or "24 below the mean" etc...
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